
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
A uniform ladder of mass  $m$  and length  $\mathcal{L}$  is resting on a wall. A man of mass  $m$  climbs up the ladder and is in perfect equilibrium with the ladder when he is  $\frac{2}{3}\mathcal{L}$  the way up the ladder. The ladder makes an angle of  $ \theta = 30^\circ $  with the horizontal floor. If the coefficient of static friction between the ladder and the wall is the same as that between the ladder and the floor, which is  $\mu$ , what is  $\mu$ , expressed to the nearest thousandth? 

*(Proposed by Ahaan Rungta)*

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