
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
The product   $$ \left(\frac{1}{2^3-1}+\frac12\right)\left(\frac{1}{3^3-1}+\frac12\right)\left(\frac{1}{4^3-1}+\frac12\right)\cdots\left(\frac{1}{100^3-1}+\frac12\right) $$   can be written as  $\frac{r}{s2^t}$  where  $r$ ,  $s$ , and  $t$  are positive integers and  $r$  and  $s$  are odd and relatively prime. Find  $r+s+t$ .

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