
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
For all primes  $p \geq 3,$  define  $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$  and  $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$ , where  $\{x\} = x - [x],$  find the value of  $f(p).$ 

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