
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
After a time $\tau$, the bottom end of the ladder moves away from the corner a distance of $x=a \tau^{2} / 2$ and gains a velocity of $v_{1}=a \tau$. At this point, the ladder will be tilted at an angle $\alpha$ to the vertical, where $\sin \alpha=a \tau^{2} /(2 l)$. Since the length of the ladder remains constant, the projections of the speeds of its ends along the direction of the ladder are identical. Denoting the speed of the top end by $v_{2}$, we obtain $v_{1} \sin \alpha = v_{2} \cos \alpha$, from which

$$
v_{2}=v_{1} \operatorname{tg} \alpha=\frac{a^{2} \tau^{3}}{\sqrt{4 l^{2}-a^{2} \tau^{4}}}
$$

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