Reverse dots and... cubes? - Snowflake edition

Question

_{A Reverse dots-and-boxes puzzle inspired by @jafe}

Albert and Robinson are playing a game of reverse dots and boxes.

• The players take turns adding one plane in one free spot on the grid (marked with light gray contours in the below image). Albert goes first.
• If a move completes a $1\times1\times1$ cube, the player gets one point and has to make another move. If two cubes are completed with a single move, the player gets two points but only has to make one additional move. The player keeps making moves until they make a move which does not complete a $1\times1\times1$ cube.
• The game ends when all possible planes have been drawn.
• Since this is a reverse game, the player with the most points loses.

Sorry for my crude drawing. In (other) words, from the origin, a stretch of 2 cubes are added to each of the six faces of the centre cube.

Which of the players can win the game played in the above grid? What strategy should they use?

_{As usual, @Deusovi got it within minutes. Kudos to him!}

Answer 1

The player who can win is

the first player

by this strategy:

Claim the end cube along five directions, then take the face of the central cube in the sixth direction. Then, they have claimed five cubes, and the other player will be forced to make moves taking all the remaining eight.