Okay. THAT. WAS. INCREDIBLE!
It took me two solid hours of work to solve and even longer to write and draw it all up here - hopefully it'll be worth it! To start us off, here is the final maze layout and routes:
In the following explanation all colours have been abbreviated to their initials as follows: G=Green, O=Orange, P=Purple, Y=Yellow.
An observation to start:
Recall that the order of moves must always be: STAIRS --> NORTH --> EAST --> SEARCH --> SOUTH --> WEST, then looping back round to STAIRS. As we start with 1 tile down, we have only 9 possible SEARCH actions before we end up in an impossible situation, unable to add another tile. This will happen on Move 58 (after 9 complete move cycles and the first 3 moves of cycle 10). This means all pieces must reach the squares by Move 57 at the very latest.
So, let’s begin... 1 tile:
Another observation: Since (i) either P or Y must take the stairs on Move 1, (ii) every piece must reach a search-space in this room in order to exit it, and (iii) pieces are not allowed to cross paths that they themselves or other pieces have already taken, it can be deduced that the room can be divided into 4 quadrants:
Every piece must leave the room through the exit within their quadrant in this diagram. The crucial move in this cycle will turn out to be Move 3 – which piece should move East? Only two pieces might be able to do so and still head in the direction of their exit: O and P. If O moves East, it will then have to perform the Search action; moreover, it will then be the only piece able to move South, which would move it away from its exit. Thus it must be P that moves East. How? It has to take the stairs first. The first few moves therefore go like this:
Move 1 (STAIRS): P.
Move 2 (NORTH): G (only option).
Move 3 (EAST): P, one space to the search-space.
Move 4 (SEARCH): P, adding tile 2, like so:
Before Move 5, think: How can we get a piece onto a search-space in 6 moves' time? It won't be Y (need Stairs-West but West comes before Stairs in the next 6 moves) or G (need West-South but South comes before West). It also can't be P (at a minimum, we need South-Stairs-West), so it must be O. How do we get O there? Either by East alone, or by South-East-North. This latter option is not possible due to the sequence of terms, so O will travel East in this run and make no other moves. Now the only piece which can go South in Move 5 is P. We end up with the following:
Move 5 (SOUTH): P, but how many spaces? Not sure yet...
Move 6 (WEST): G (only option).
Move 7 (STAIRS): Either Y or P (if P has moved one or three spaces South).
Move 8 (NORTH): The only piece that could go North is P, but only if it took the stairs in Move 7. Therefore, after Move 8 Y is still on its starting spot and P is in one of three possible spots (since all of these can be reached via the bottom stairs, let's assume for now that that's what P did - we can always correct it later if we need the space back...).
Move 9 (EAST): O (as explained above).
Move 10 (SEARCH): O, adding tile 3, like so:
Move 11 (SOUTH): G (only option).
Move 12 (WEST): P (only option) - but we still don't know which of 3 spots P is in.
Move 13 (STAIRS): Y (only option).
Move 14 (NORTH): P is the only option, but only if it only went North by just one space in Move 8. Since we must avoid leading P into a dead end, P must now be on the search-space.
Move 15 (EAST): O (only option) - but we cannot be sure yet by how many spaces...
Move 16 (SEARCH): We have two choices: G or P. To work out which, we need to look forward in the tile deck at the entry and exit positions on the remaining tiles. Ultimately we will see that this means we cannot go with P here. Consider: if we place Tile 4 next to P, either G or O needs to get to one of the squares in the bottom row of this tile. The wall along the tile top means that G would need to go around this tile, requiring a 'straight-through' followed by two 'left turns' - this would require the use of Tiles 5, 6 and 7 for G, which will be impossible to achieve since Tile 5 cannot be crossed in one move cycle, given the sequence that the moves must follow. Alternatively, O would need to navigate around the already-placed Tile 2, which would require 'straight through', 'right turn', straight through', right turn' - but from Tile 8 onwards (at which point a 'right turn' would be required), all pieces have 'straight through' structure. Thus in Move 16 we have to search with G, by contradiction. Tile 4 goes on, like so:
Move 17 (SOUTH): O is the only option - but where is it? In order to end up on the triangle and have a free escape route out of the room when needing to pursue the square afterwards, O must be within the central 4 squares of the room. Why not take the shortest route for now and say it is directly above the triangle - that way we now have O ON ITS TRIANGLE. (We will confirm later whether we were correct to do this - see Move 33...)
Move 18 (WEST): We need a piece on the stairs for the next move - move G West. Which staircase to choose (we have two options)? Well, stopping at the first will result in a dead-end after using them, so bypass those and head three spaces West to the second.
Move 19 (STAIRS): G (only option).
Move 20 (NORTH): G (only option). We're going to need to head towards those stairs in the middle for the next cycle if we want to keep the game alive, so move two or three spaces North (one is not an option, since there is no further North move before the next staircase...).
Move 21 (EAST): G (only option, since O must stay on its triangle until all pieces are on their triangles) - two possible positions still.
Move 22 (SEARCH): P, adding tile 5, like so:
Move 23 (SOUTH): The only piece which could move South is G, but only if it is currently at the top of its room. This finalises Moves 20 and 21.
Move 24 (WEST): Either Y or P goes West. We need someone on a search-space by the end of this move-set, so move Y.
Move 25 (STAIRS): G (only option).
Move 26 (NORTH): G (only option).
Move 27 (EAST): G (only option).
Move 28 (SEARCH): Y, adding tile 6, like so:
Move 29 (SOUTH): G (only option) - now ON ITS TRIANGLE.
Move 30 (WEST): P (only option).
Move 31 (STAIRS): P (only option - G must stay on its triangle until all others are in position) - now ON ITS TRIANGLE.
Move 32 (NORTH): Y (only option) - now ON ITS TRIANGLE.
So, if we were correct to move O to its triangle in Move 17, we have at this point successfully positioned all counters on their triangles - halfway there! Now to get to those pesky squares... (Note that we can now confirm that we were correct to move O to its triangle in Move 17 - for Move 34 we will need to have a piece on a search-space; since Move 33 is a move East, the only piece that can successfully move East from their triangle to a search-space is O. If in Move 17 we had instead moved O South to the space one to the left of its triangle, Move 33 would need to be used to get it onto its triangle, no piece would then be on a search-space just before Move 34 and our strategy would collapse altogether. Therefore, our Move 17 above is the correct choice...).
Move 33 (EAST): O (only option, since we need to reach a search-square now).
Move 34 (SEARCH): O, adding tile 7, like so:
At this point, note again that the tiles still to be placed all have the 'straight through' structure (i.e the entry and exit are on opposing walls). This means these pieces can never be used as a way to get from the current positions to one of the four squares, which are all already out in the maze. These tiles will therefore never be entered by a counter - they are purely dead-end pieces which will need to be laid through the search-square moves.
As a result of this, we do not want to move O South at this point, as it will never end up getting to a square if we do! This leaves one viable option for the next move...
Move 35 (SOUTH): P (only option).
Move 36 (WEST): P (only option), but only if P moved only one space in Move 35 (or it will have ended up on a staircase).
Move 37 (STAIRS): P - although G is an option, we need a counter on a search-space for the Search action in a few moves' time...
Move 38 (NORTH): Y or O. Which one? Well, in Move 39 one of these two pieces will need to move East. If O moves North now, that will be impossible as both would have a wall to their right. Thus Y must move North now.
Move 39 (EAST): Either Y or O - we can't be sure which just yet...
Move 40 (SEARCH): P, adding tile 8, like so:
Before Move 41 we need to ask ourselves: Who can move West in Move 42? Well, G and Y can't, and we don't want P to go West - that way leads to a dead-end. So O is our only option. That means that on Move 41, O must move South from its current position... which means that O (not Y) must have moved East in Move 39. In order to be able to move West in Move 42, O must be at the far end of its tile, giving us the following sequence of moves:
Move 41 (SOUTH): O - but by one or two spaces? Not sure yet...
Move 42 (WEST): O again - but do we send it to its square or to the bottom row of the tile? Again, not sure yet...
Move 43 (STAIRS): G (only option) - now ON ITS SQUARE.
Considering the moves remaining to us before the next Search action must be carried out, only O could realistically do it - and only if we have not sent it to its square in Move 42, tempting as it was... Therefore, O is on the bottom row of its tile and must already be sitting on the search-square. We do not want to move it North and away from this square just yet, so instead we have:
Move 44 (NORTH): P (only viable option).
Move 45 (EAST): P or Y (can't be sure which yet).
Move 46 (SEARCH): O, adding tile 9, like so:
Note at this point that G is home and we do not want O or P to go South any more. This means only Y must be allowed to travel South from now on, which in turn means that in Move 45 we must have moved Y to a position from which it may move South in Move 47. The moves proceed as follows:
Move 47 (SOUTH): Y (only option).
Move 48 (WEST): O (only option).
Move 49 (STAIRS): Y (only option).
Move 50 (NORTH): O (only option).
Move 51 (EAST): P (we need a counter on the stairs for the next cycle).
Move 52 (SEARCH): Y, adding the final tile 10, like so:
(Note that although its South wall blocks off a search-space on an older tile, this is a perfectly legitimate tile placement by the rules of the game as specified in the question.)
Now it's the home straight! Five moves left, all vital:
Move 53 (SOUTH): Y.
Move 54 (WEST): Y - now ON ITS SQUARE.
Move 55 (STAIRS): P.
Move 56 (NORTH): P - now ON ITS SQUARE.
Move 57 (EAST): O - now ON ITS SQUARE.
HOORAY! All pieces are now on their squares, the puzzle is solved, there is nothing further to do and I need to go get a drink and rock quietly backwards and forwards to myself in a corner somewhere! Phew! Epic puzzle – huge congrats to the setter for this truly fantastic accomplishment!