
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 solutions to the equation $z^{2}+z+1=0$. <br/>$(1)$ Find the value of $\frac{1}{z_1}+\frac{1}{z_2}$; <br/>$(2)$ If the point representing $z_{1}$ in the complex plane is in the third quadrant, and $z_{1}\cdot \left(a+i\right)$ is a purely imaginary number, where $a\in R$, find the value of $a$.

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