
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]


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A student adds up rational fractions incorrectly:

\[\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y}\quad (\star) \]

Despite that, he sometimes obtains correct results. For a given fraction  $\frac{a}{b},a,b\in\mathbb{Z},b>0$ , find all fractions  $\frac{x}{y},x,y\in\mathbb{Z},y>0$  such that the result obtained by  $(\star)$  is correct.

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