
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
Let  $S$  be the number of bijective functions  $f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\}$  such that  $f((m+n)\pmod{17})$  is divisible by  $17$  if and only if  $f(m)+f(n)$  is divisible by  $17$ . Compute the largest positive integer  $n$  such that  $2^n$  divides  $S$ .

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