
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) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.

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