
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
Compute the smallest positive integer \( n \) for which there are at least two odd primes \( p \) such that
$$
\sum_{k=1}^{n} (-1)^{\nu_{p}(k!)} < 0.
$$

Note: For a prime \( p \) and a positive integer \( m \), \( \nu_p(m) \) is the exponent of the largest power of \( p \) that divides \( m \); for example, \( \nu_3(18) = 2 \).

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