
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
Define \( n! = 1 \times 2 \times \ldots \times n \). For example, \( 5! = 1 \times 2 \times 3 \times 4 \times 5 \). If \( \frac{n! \times (n+1)!}{2} \) (where \( n \) is a positive integer and \( 1 \leq n \leq 100 \)) is a perfect square, for example, when \( n=7 \):
\[ \frac{n! \times (n+1)!}{2} = \frac{7 \times 8!}{2} = \frac{7 \times 8!}{2} = \frac{7 \times (7 \times 8)}{2} = (7!)^2 \times 4 = (7!)^2 \times 2^2 \]
is a perfect square. Find the sum of all such \( n \) that satisfy this condition.

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