
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
The polynomial  $A(x) = x^2 + ax + b$  with integer coefficients has the following property: 
for each prime  $p$  there is an integer  $k$  such that  $A(k)$  and  $A(k + 1)$  are both divisible by  $p$ . 
Proof that there is an integer  $m$  such that  $A(m) = A(m + 1) = 0$ .

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