
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 \(x^{2} + 5bx + c = 0\) has roots \(x_{1}\) and \(x_{2}\), where \(x_{1} \neq x_{2}\), and a certain number is a root of the equation \(y^{2} + 2x_{1}y + 2x_{2} = 0\) and a root of the equation \(z^{2} + 2x_{2}z + 2x_{1} = 0\). Find \(b\).

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