
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 fully set up, the top of the tripod is located right above the centroid of an equilateral triangle formed by the legs on the ground and is 5 feet from the ground. Due to damage, the bottom 1 foot of one leg breaks off. Denoting $h$ as the height of the tripod's top from the ground after setting it up with the damaged leg, find $h$ and determine $\lfloor m+\sqrt{n}\rfloor$ where $h$ is in the form $\frac m{\sqrt{n}}$, with $m$ and $n$ as positive integers and $n$ not being divisible by any prime squared.

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