
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  $f$  be a random permutation on  $\{1, 2, \dots, 100\}$  satisfying  $f(1) > f(4)$  and  $f(9)>f(16)$ . The probability that  $f(1)>f(16)>f(25)$  can be written as  $\frac mn$  where  $m$  and  $n$  are relatively prime positive integers. Compute  $100m+n$ .

Note: In other words,  $f$  is a function such that  $\{f(1), f(2), \ldots, f(100)\}$  is a permutation of  $\{1,2, \ldots, 100\}$ .

*Proposed by Evan Chen*

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