
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
Calculate the integral

$$
\int_{C}(1+i-2 \bar{z}) d z
$$

along the paths connecting the points \( z_{1}=0 \) and \( z_{2}=1+i \):

1. Along a straight line.
2. Along the parabola \( y=x^{2} \).
3. Along the broken line \( z_{1} \rightarrow z_{3} \rightarrow z_{2} \), where \( z_{3}=1 \).

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