
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
On the edge $BB_{1}$ of a regular triangular prism $ABC A_{1} B_{1} C_{1}$, a point $T$ is taken such that $BT: B_{1} T=2:3$. Point $T$ is the vertex of a right circular cone, such that three vertices of the prism belong to the circle of its base.

a) Find the ratio of the height of the prism to the edge of its base.

b) Let it be additionally known that $C C_{1}=5$. Find the volume of the cone.

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