
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 integers \(a, b, c\), and \(d\) are each equal to one of \(1, 2, 3, 4, 5, 6, 7, 8, 9\). (It is possible that two or more of the integers \(a, b, c\), and \(d\) have the same value.) The integer \(P\) equals the product of \(a, b, c\), and \(d\); that is, \(P = abcd\).

(a) Determine whether or not \(P\) can be a multiple of 216.

(b) Determine whether or not \(P\) can be a multiple of 2000.

(c) Determine the number of possible different values of \(P\) that are divisible by 128 but are not divisible by 1024.

(d) Determine the number of ordered quadruples \((a, b, c, d)\) for which \(P\) is 98 less than a multiple of 100.

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