
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
Given that \( z_{1}, z_{2}, \cdots, z_{5} \) are the five complex roots of the equation
\[ x^{5} + 2x^{4} + 3x^{3} + 4x^{2} + 5x + 6 = 0, \]
find the value of \( \sum_{i=1}^{5} \frac{z_{i}}{z_{i}^{2} + 1} \).

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