
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
Consider quadrangular pyramids \( M A B C D \) with the following properties: the base of the pyramid is a convex quadrilateral \( A B C D \), where \( A B = B C = 1 \) and \( C D = D A = 2 \), and each of the planes of the lateral faces \( M A B, M B C, M C D, M D A \) forms an angle of \( 45^{\circ} \) with the plane of the base.

a) Find the volume of such a pyramid if its height, dropped from the vertex \( M \), is \( \frac{9}{5} \).

b) For what height is the volume of the discussed pyramids maximized, and what is this maximum volume?

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