
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
Find a particular solution $u(x, t)$ of the differential equation $\frac{\partial^{2} u}{\partial t^{2}} - a^{2} \frac{\partial^{2} u}{\partial x^{2}} = 0$, satisfying the boundary conditions:

1. $u(0, t) = 0$
2. $u(l, t) = 0$
3. $u(x, 0) = \varphi_1(x)$
4. $u_t(x, 0) = \varphi_2(x)$.

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