
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
Ten lockers are in a row. The lockers are numbered in order with the positive integers 1 to 10. Each locker is to be painted either blue, red, or green subject to the following rules:

- Two lockers numbered \( m \) and \( n \) are painted different colors whenever \( m - n \) is odd.
- It is not required that all 3 colors be used.

In how many ways can the collection of lockers be painted?

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