
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 cross sections of a right pentagonal pyramid are obtained by slicing the pyramid with planes parallel to the pentagonal base. The areas of these cross sections are $100\sqrt{3}$ square feet and $225\sqrt{3}$ square feet, respectively. The two planes are $5$ feet apart. Determine the distance from the apex of the pyramid to the larger cross section.

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