
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  $ p>2 $  be a prime number. For any permutation  $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $  of the set  $ S = \{ 1, 2, \cdots , p \} $ , let  $ f( \pi ) $  denote the number of multiples of  $ p $  among the following  $ p $  numbers:
\[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p)  \]
Determine the average value of  $ f( \pi) $  taken over all permutations  $ \pi $  of  $ S $ .

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