
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  $O$  be the set of odd numbers between 0 and 100.  Let  $T$  be the set of subsets of  $O$  of size  $25$ .  For any finite subset of integers  $S$ , let  $P(S)$  be the product of the elements of  $S$ .  Define  $n=\textstyle{\sum_{S \in T}} P(S)$ .  If you divide  $n$  by 17, what is the remainder?

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