
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 trivet has three legs, two of which are each 7 feet long and one leg, which is shorter due to a defect, is 6 feet long. When the trivet is set up, the angle between any two longer legs is 60 degrees, whereas the angles involving the shorter leg with each of the longer legs is 75 degrees. During the setup, the shorter leg breaks off an additional 1 foot from its lower end. Determine the height $h$ of the top of the trivet from the ground under these new conditions.

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