
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
The cells of a  $5\times5$  grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through  $9$  cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly  $3$  red cells, exactly  $3$  white cells, and exactly  $3$  blue cells no matter which route he takes.

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