
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
Given a regular triangular pyramid $S-ABC$, with one base of a regular triangular prism having its three vertices on the edges of the pyramid and the other base on the base of the regular triangular pyramid. If the height of the regular triangular pyramid is $15cm$, the side length of the base is $12cm$, and the lateral area of the inscribed regular triangular prism is $120cm^{2}$, find:<br/>
$(1)$ the height of the regular triangular prism;<br/>
$(2)$ the ratio of the lateral area of the small pyramid cut by the base of the prism to the lateral area of the original pyramid.

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