
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  $(x - a)(x - 10) + 1$  factors in the product  $(x + b)(x + c)$  with integers  $b$  and  $c$ .
b) Find nonzero and nonequal integers  $a, b, c$  so that  $x(x - a)(x - b)(x - c) + 1$  factors into the product of two polynomials with integer coefficients.

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