
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 respectively on the lines $y= \frac{\sqrt{2}}{2}x$ and $y=-\frac{\sqrt{2}}{2}x$, with $|AB|=2\sqrt{2}$. Let O be the origin, and the moving point P satisfies $\overrightarrow{OP} = \overrightarrow{OA} + \overrightarrow{OB}$.
(Ⅰ) Find the equation of the trajectory $C_1$ of the moving point P;
(Ⅱ) A hyperbola $C_2$ has its upper vertex as one of the foci, and the equation for one of its asymptotes is $x+2y=0$. Find the equation of the hyperbola $C_2$.

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