
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
One cylinder has a radius of \(\sqrt{2}\) and a height of \(a\). Another cylinder has a radius of \(\sqrt{5}\) and a height of \(b\). How many pairs of positive integers \((a, b)\) are there so that the sum of the volumes of the two cylinders is \(10 \pi t\)?

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