
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
In the complex plane, the complex number \( z_{1} \) moves along the line segment connecting \( 1+2i \) and \( 1-2i \), and the complex number \( z_{2} \) moves along the circumference of a circle centered at the origin with radius 1. What is the area of the shape formed by the trajectory of \( z_{1} + z_{2} \)?

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