
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  $f(x)$  be the polynomial  $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$ .  Let  $c$  be the coefficient of  $x^{48}$  in  $f(x)$ .  When  $c$  is divided by 101, what is the remainder?  (The remainder is an integer between 0 and 100.)

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