
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 of all elements in a finite set $S$ is referred to as the "product number" of set $S$. Given the set $M=$ $\left\{\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \cdots, \frac{1}{100}\right\}$, determine the sum of the "product numbers" of all even-numbered subsets (2 elements, 4 elements, $\cdots$, 98 elements) of $M$.

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