
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 \(a, b \in \mathbf{R}\). Given that the equation \(\left(z^{2}+a z+b\right)\left(z^{2}+a z+2 b\right)=0\) has 4 distinct complex roots \(z_{1}, z_{2}, z_{3}, z_{4}\), and these roots correspond to the vertices of a square with side length 1 in the complex plane, 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}
user