
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 ordered pairs \((a, b)\) of positive integers satisfy the equation
\[
a \cdot b + 97 = 18 \cdot \text{lcm}(a, b) + 14 \cdot \text{gcd}(a, b),
\]
where \( \text{gcd}(a, b) \) denotes the greatest common divisor of \(a\) and \(b\), and \( \text{lcm}(a, b) \) denotes their least common multiple?
A) 0
B) 2
C) 4
D) 6
E) 8

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