
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 \( \mathbf{n} \) is a positive integer and \( 1 \leq n \leq 100 \)) is a perfect square, what is the sum of all such \( \mathbf{n} \)?

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