
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
If \( a * b = ab + 1 \), and \( s = (2 * 4) * 2 \), find \( s \).

If the \( n \)-th prime number is \( s \), find \( n \).

If \( K = \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right) \cdots \left(1 - \frac{1}{50}\right) \), find \( K \) in the simplest fractional form.

If \( A \) is the area of a square inscribed in a circle of radius 10, find \( A \).

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