
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
For all positive integers  $n$ , denote by  $\sigma(n)$  the sum of the positive divisors of  $n$  and  $\nu_p(n)$  the largest power of  $p$  which divides  $n$ .  Compute the largest positive integer  $k$  such that  $5^k$  divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where  $N=6^{1999}$ .

*Proposed by David Altizio*

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