
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 the same, and the top of the tripod is $5$ feet from the ground. Suppose that the lower $2$ feet of one leg break off. Let \( h \) be the 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 \( \frac{m}{\sqrt{n}} \), where \( m \) and \( n \) are positive integers and \( n \) is not divisible by the square of any prime. Find \( \lfloor m + \sqrt{n} \rfloor \). (The notation \( \lfloor x \rfloor \) denotes the greatest integer that is less than or equal to \( x \).)

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