
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
a) Find an integer \( a \) for which \( (x-a) \times (x-10) + 1 \) can be factored into the product \( (x+b)(x+c) \) of two binomials with integer \( b \) and \( c \).

b) Find distinct non-zero integers \( a, b, \) and \( c \) such that the expression \( x(x-a)(x-b) \times (x-c) + 1 \) can be factored into the product of two polynomials with integer coefficients.

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