
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
There is a product $a_{1} \cdot a_{2} \cdot \cdots \cdot a_{100}$ written on the blackboard, where $a_{1}, a_{2}, \cdots, a_{100}$ are positive integers. If one of the multiplication symbols is changed to an addition symbol (while keeping the rest multiplication symbols), it is found that 32 out of the 99 resultant sums are even. How many even numbers can there be at most among $a_{1}, a_{2}, \cdots, a_{100}$?

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