
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 cone has a uniform circular base of radius $6$ feet and a slant height of $13$ feet. The cone’s side breaks, causing the height from the base to the tip along the cone's side to reduce by $2$ feet while maintaining a right angle at the vertex. Determine the new height, $H$, of the cone from the base to the tip.

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