
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
Let $A$ and $B$ be moving points on the lines $y= \frac{ \sqrt{2}}{2}x$ and $y=- \frac{ \sqrt{2}}{2}x$, respectively, with $|AB|=2 \sqrt{2}$. Let $O$ be the coordinate origin, and let moving point $P$ satisfy $\overrightarrow{OP}= \overrightarrow{OA}+ \overrightarrow{OB}$.
(I) Find the trajectory equation $C_{1}$ of moving point $P$.
(II) A hyperbola $C_{2}$ has one focus at the upper vertex of $C_{1}$, and one asymptote with the equation $x+2y=0$. Find the equation of hyperbola $C_{2}$.

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