
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 first mixed boundary value problem for the heat equation on the interval:

$$
\begin{aligned}
& u_{t}=u_{x x}, \quad x \in(0,1), \quad t \in(0, \infty) \\
& u(x, 0)=\sin ^{3} 2 \pi x, \quad x \in(0,1) \\
& u(0, t)=u(1, t)=0, \quad t \in(0, \infty)
\end{aligned}
$$

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