
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 \( m \) be the product of all positive integer divisors of 360,000. Suppose the prime factors of \( m \) are \( p_{1}, p_{2}, \ldots, p_{k} \), for some positive integer \( k \), and \( m = p_{1}^{e_{1}} p_{2}^{e_{2}} \cdot \ldots \cdot p_{k}^{e_{k}} \), for some positive integers \( e_{1}, e_{2}, \ldots, e_{k} \). Find \( e_{1} + e_{2} + \ldots + e_{k} \).

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