
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
Consider a polynomial given by:
$$p(x) = a(x-1)^4 + b(x-1)^3 + c(x-1)^2 + b(x-1) + a = 0,$$
where $a, b, c$ are integers with no common prime factors. Determine $|c|$ if $x=1+i$ is a root of $p(x)$.

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