
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
In an alley of width $w$, a ladder rests against two walls. At one end, say point $Q$, the ladder makes a $30^\circ$ angle with the ground, and on the opposite wall at $R$, it makes a $60^\circ$ angle with the ground. Given that the height at point $Q$ is $h$ and at $R$ is $k$, find the length $a$ of the ladder.
A) $h$
B) $\sqrt{3}h$
C) $2h$
D) $\frac{h}{2}$
E) $\frac{h}{\sqrt{3}}$

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