
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
Two planes parallel to the hexagonal base of a right hexagonal pyramid cut through it, producing two cross-sections with areas of $96\sqrt{3}$ square feet and $216\sqrt{3}$ square feet. The distance between these two planes is $12$ feet. The lateral edges of the pyramid are perpendicular to its base. Determine how far the larger cross section is from the apex of the pyramid, in feet.

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