
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
It is known that the equation \(x^2 + 5b x + c = 0\) has roots \(x_1\) and \(x_2\), where \(x_1 \neq x_2\), and a certain number is a root of both equations \(y^2 + 2x_1 y + 2x_2 = 0\) and \(z^2 + 2x_2 z + 2x_1 = 0\). Find \(b\).

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