
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 base of the quadrilateral pyramid $M A B C D$ is a rectangle $A B C D$. The faces $A B M$ and $B C M$ of the pyramid are perpendicular to the base. The faces $M C D$ and $M A D$ form dihedral angles $\beta$ and $\alpha$ with the base, respectively.

Find the volume of the pyramid if $M D = 3$.

For what values of $\alpha$ and $\beta$ will the volume of the pyramid be maximized?

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