
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  $N{}$  be a fixed positive integer,  $S{}$  be the set  $\{1, 2,\ldots , N\}$  and  $\mathcal{F}$  be the set of functions  $f:S\to S$  such that  $f(i)\geqslant i$  for all  $i\in S.$  For each  $f\in\mathcal{F}$  let  $P_f$  be the unique polynomial of degree less than  $N{}$  satisfying  $P_f(i) = f(i)$  for all  $i\in S.$  If  $f{}$  is chosen uniformly at random from  $\mathcal{F}$  determine the expected value of  $P_f'(0)$  where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\]*Proposed by Ishan Nath*


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