
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
We know the total surface area of a right circular cone is $\pi m^{2}$ and its volume is $\frac{1}{3} \pi a^{3}$. Find its height and the radius of its base. When the volume is constant, when will the surface area be minimized? If the surface area is constant, when will the volume be maximized? In this latter case, compute the angle formed by the cone's slant height and its axis.

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