
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 consider quadrilateral pyramids \(K A B C D\) with the following properties: the base of the pyramid is a convex quadrilateral \(A B C D\) where \(A B = B C = 3\), \(C D = D A = 4\), and each of the planes of the lateral faces \(K A B\), \(K B C\), \(K C D\), \(K D A\) 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 vertex \(K\) is 2.

b) For what height is the volume of the considered pyramids maximal and what is this volume?

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