
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
Let \(\pi\) be a uniformly random permutation of the set \(\{1,2, \ldots, 100\}\). The probability that \(\pi^{20}(20) = 20\) and \(\pi^{21}(21) = 21\) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a + b\). (Here, \(\pi^{k}\) means \(\pi\) iterated \(k\) times.)

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