
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
An ordered pair \((a, b)\) of positive integers is called spicy if \(\operatorname{gcd}(a+b, ab+1)=1\). Compute the probability that both \((99, n)\) and \((101, n)\) are spicy when \(n\) is chosen from \(\{1, 2, \ldots, 2024\}\) uniformly at random.

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