
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 complex numbers $z_1 = m + ni$, $z_2 = 2 - 2i$ and $z = x + yi$, where $z = \overline{z_1}i - z_2$, and $m$, $n$, $x$, $y \in \mathbb{R}$. If the corresponding point $M(m, n)$ of complex number $z_1$ moves on the curve $y = \frac{1}{2}(x + 2)^2 + \frac{5}{2}$, find the equation of the trajectory $C$ of the corresponding point $P(x, y)$ of the complex number $z$.

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