
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 Laplace's equation in a cylinder

$$
\begin{array}{ll}
\Delta u=0, \quad 0 \leqslant r<1, & 0<z<1 \\
\left.u\right|_{z=0}=1-r^{2}, & 0 \leqslant r<1 \\
\left.u\right|_{z=1}=0, & 0 \leqslant r<1 \\
\left.u\right|_{r=1}=0, & 0<z<1 .
\end{array}
$$

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