
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
How many different pairs of positive integers \( (a, b) \) are there such that \( a + b \leq 100 \) and \( \frac{a + \frac{1}{b}}{\frac{1}{a} + b} = 13 \)?
(a) 1
(b) 5
(c) 7
(d) 9
(e) 13

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