
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 $T A B C D$ with the following properties: the base of the pyramid is a convex quadrilateral $A B C D$, in which $A B=B C=1$, $C D=D A=2$, and each of the planes of the lateral faces $T A B, T B C, T C D, T D A$ forms an angle of $60^{\circ}$ with the plane of the base.

a) Find the volume of such a pyramid if its height, dropped from the vertex $T$, is 2.

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

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