
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
Determine the number of integers \( n \) such that the product \( Q(n)R(n) \leq 0 \), where \( Q(x) = (x - 1^2)(x - 2^2) \cdots (x - 50^2) \) and \( R(x) = (x - 0)(x - 1)(x - 2) \cdots (x - 50) \).

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