
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
15. Let  $a_{n}$  denote the number of ternary strings of length  $n$  so that there does not exist a  $k<n$  such that the first  $k$  digits of the string equals the last  $k$  digits. What is the largest integer  $m$  such that  $3^{m} \mid a_{2023}$  ?

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