
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
The bases of a truncated pyramid are regular triangles \(ABC\) and \(A_1B_1C_1\) with side lengths 3 and 2, respectively. The segment connecting vertex \(C_1\) with the center \(O\) of the base \(ABC\) is perpendicular to the bases and \(|C_1 O|=3\). A plane is drawn through vertex \(B\) and points \(M\) and \(N\), which are the midpoints of edges \(A_1B_1\) and \(B_1C_1\). Cylinders located inside the polyhedron \(ABC A_1MNC_1\), with bases in the face \(A_1MNC_1\), are considered. Find: a) the maximum volume of such cylinders with a given height \(h\); b) the maximum volume among all considered cylinders.

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