
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 complex numbers $z_{1}$ and $z_{2}$ are two roots of the equation $x^{2}+x+1=0$, find $\frac{{z_{2}}}{{z_{1}+1}}+\frac{{z_{1}}}{{z_{2}+1}}$.

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