
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
Find all value of  $ n$  for which there are nonzero real numbers  $ a, b, c, d$  such that after expanding and collecting similar terms, the polynomial  $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$  has exactly  $ n$  nonzero coefficients.

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