
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
Let \(a, b \in \mathbb{R}\). If the equation \(\left(z^2 + az + b\right)\left(z^2 + az + 2b\right) = 0\) has four distinct complex roots \(z_1, z_2, z_3, z_4\), and the points corresponding to these roots on the complex plane are exactly the four vertices of a square with side length 1, find the value of \(\left|z_1\right| + \left|z_2\right| + \left|z_3\right| + \left|z_4\right|\).

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