
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 8 feet. When the tripod is set up, the angle between any pair of legs is equal, and the top of the tripod is 6 feet from the ground level. After setting up, the lower 2 feet of one leg sinks into a soft ground. If the ground around the other two legs remains firm, let \( h \) be the height in feet of the top of the tripod from the ground level. Assume the ground slope where the leg sank does not affect the other two legs. Find \( h \) in the form \(\frac{m}{\sqrt{n}}\), where \( m \) and \( n \) are positive integers and \( n \) is not divisible by the square of any prime. Calculate \(\lfloor m+\sqrt{n} \rfloor\).

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