
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
In the diagram, \(P A B C D\) is a pyramid with square base \(A B C D\) and with \(P A = P B = P C = P D\). Suppose that \(M\) is the midpoint of \(P C\) and that \(\angle B M D = 90^{\circ}\). Triangular-based pyramid \(M B C D\) is removed by cutting along the triangle defined by the points \(M, B\), and \(D\). The volume of the remaining solid \(P A B M D\) is 288. What is the length of \(A B\)?

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