
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
Consider a product \(a_{1} a_{2} \ldots a_{100}\) written on a board, where \(a_{1}, \ldots, a_{100}\) are natural numbers. We consider 99 expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that exactly 32 of these expressions are even. What is the maximum number of even numbers among \(a_{1}, a_{2}, \ldots, a_{100}\) that could exist?

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