
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 which natural numbers $n$ is the number 

$$
\frac{1!\cdot 2!\cdot 3!\cdot \ldots \cdot(2n)!}{(n+1)!}
$$

a perfect square? (As usual, $n!$ denotes the product of all natural numbers up to $n$. For example, $4!=1 \cdot 2 \cdot 3 \cdot 4$.)

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