
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
308. a) Find an integer \( a \) such that

\[ (x-a)(x-10) + 1 \]

can be factored into the product \((x+b)(x+c)\) of two factors with integer \( b \) and \( c \).

b) Find such non-zero, distinct integers \( a, b, c \) so that the fourth-degree polynomial with integer coefficients

\[ x(x-a)(x-b)(x-c) + 1 \]

can be represented as the product of two other polynomials with integer coefficients.

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