
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
Is there a real number $m$ such that the equations $x^2 + mx + 2 = 0$ and $x^2 + 2x + m = 0$ have exactly one common real root? If such a real number exists, find this $m$ and the common real root of the two equations; if it does not exist, explain why.

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