
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
A permutation  $(a_1, a_2, a_3, \dots, a_{100})$  of  $(1, 2, 3, \dots, 100)$  is chosen at random. Denote by  $p$  the probability that  $a_{2i} > a_{2i - 1}$  for all  $i \in \{1, 2, 3, \dots, 50\}$ . Compute the number of ordered pairs of positive integers  $(a, b)$  satisfying  $\textstyle\frac{1}{a^b} = p$ .

*Proposed by Aaron Lin*

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