
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 the parabola $C$: 
$$
\begin{cases} 
\left.\begin{matrix}
x=2t^{2} \\ 
y=2t
\end{matrix}\right.
\end{cases}
$$
(where $t$ is the parameter), let $O$ be the origin, and point $M(x_{0}, y_{0})$ moves on $C$. If point $P(x, y)$ is the midpoint of segment $OM$, then the general equation of the trajectory of point $P$ is \_\_\_\_\_\_.

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