
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 \( a_{1}, a_{2}, \cdots, a_{k}\left(k \in \mathbf{Z}_{+}\right) \) be integers greater than 1, and they satisfy
\[ 
\left(a_{1}!\right)\left(a_{2}!\right) \cdots\left(a_{k}!\right) \mid 2017!
\]

Determine the maximum value of \( \sum_{i=1}^{k} a_{i} \) as \( k \) varies.

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