
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
In a right prism with triangular bases, each base triangle is an isosceles triangle with two sides of length 3 and an included angle of \(60^\circ\). Given the sum of the areas of three mutually adjacent faces (two lateral faces and one base) is 30, determine the height of the prism that maximizes its volume.

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