
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
Solve the Dirichlet boundary value problem for the Poisson equation in a ring:

$$
\begin{gathered}
\Delta u = r^{3} \cos \varphi, \quad 1 < r < 2, \\
u|_{r=1} = \cos 2 \varphi, \\
\left.\frac{\partial u}{\partial r}\right|_{r=2} = \sin 3 \varphi.
\end{gathered}
$$

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