
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 tripod has three legs each of length $6$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is initially $5$ feet from the ground. After an accident, the lower $1.5$ feet of one leg breaks off. Let $h$ be the new height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac{p}{\sqrt{q}}$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Find $\lfloor p+\sqrt{q}\rfloor.$

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