
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Each unit square of a  $4 \times 4$  square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)

[asy]
draw((0,0) -- (2,0) -- (2,1) -- (0,1));
draw((0,0) -- (0,2) -- (1,2) -- (1,0));
draw((4,1) -- (6,1) -- (6,2) -- (4,2));
draw((4,2) -- (4,0) -- (5,0) -- (5,2));
draw((10,0) -- (8,0) -- (8,1) -- (10,1));
draw((9,0) -- (9,2) -- (10,2) -- (10,0));
draw((14,1) -- (12,1) -- (12,2) -- (14,2));
draw((13,2) -- (13,0) -- (14,0) -- (14,2));
[/asy]

*Proposed by Andrew Lin.*

Present the answer in LaTex format: \boxed{Your answer}
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