
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
Convert to polar coordinates and compute the double integrals.

a) $\iint_{\Omega} x y^{2} \, dx \, dy$, where the region $\Omega$ is bounded by the circles $x^{2}+(y-1)^{2}=1$ and $x^{2}+y^{2}=4y$.

b) $\iint_{\Omega} e^{-x^{2}-y^{2}} \, dx \, dy$, where $\Omega$ is the circle $x^{2}+y^{2} \leqslant R^{2}$.

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