
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
There exist integers \( a, \) \( b, \) and \( c \) such that
\[ (x - a)(x - 5) + 1 = (x + b)(x + c). \]
Find all possible values of \( a, \) considering two scenarios where \( (b + 5)(c + 5) = 1 \) and \( (b + 5)(c + 5) = 4 \).

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