What's the phrase I'm looking for?
Note: The specific colors I've chosen is not important.
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The phrase you're looking for is:
To solve, first recognise that what we have here is:
an inwardly spiralling trail beginning at the red '1' space, passing through blue, then dark red, dark grey, yellow, cyan, etc. There are twelve sets of colours, and if we enter one letter in each it is possible to enter the names of the twelve signs of the Zodiac into this spiral, preserving the order of the signs like so:
Now all that remains is to:
read off the letters indicated by numbers in that numerical order. This spells out the phrase ASTROLOGICAL SIGNS (which suggests to me that the missing six-letter word in the title - which conjures up the idea of looking up at the stars - is 'Zodiac'...).
The diagram reproduced as text: