
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 $\theta$ is an interior angle of a triangle, let $z=\cos \theta +i\sin \theta$, and it satisfies $|z+1|=1$.  
$(1)$ Find $1+z+z^{2}$;  
$(2)$ Let $z$, $-2\overline{z}$, and $1+z+z^{2}$ correspond to points $A$, $B$, and $C$ on the complex plane, respectively. Find the area of $\triangle ABC$.

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