
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 integer triples \((a, b, c)\) such that \(0 < |a| < |b| < |c|\), and the polynomial 
$$
f(x) = x(x - a)(x - b)(x - c) + 1
$$
is reducible in \(\mathbf{Z}[x]\) (i.e., can be factored into the product of two polynomials with integer coefficients and lower degrees).

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