
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 \( p = p_{1} p_{2} \ldots p_{6} \) be a permutation of the integers from 1 to 6. For any such permutation \( p \), we count how many integers there are which have nothing bigger on their left. We define \( f(p) \) as the number of these integers in the permutation \( p \). For example, \( f(612345) = 1 \) because only 6 has no larger integer to its left. On the other hand, \( f(135462) = 4 \) because only 1, 3, 5, and 6 meet the condition. 

Let \( S \) be the sum of \( f(p) \) over all \( 6! \) different permutations. Find the sum of the digits of \( S \).

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