
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
A  $1 \times n$  rectangle ( $n \geq 1 $ ) is divided into  $n$  unit ( $1 \times 1$ ) squares. Each square of this rectangle is colored red, blue or green. Let  $f(n)$  be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of  $f(9)/f(3)$ ? (The number of red squares can be zero.)

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