
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
In a Cartesian coordinate system, there are two fixed points $F_1$ and $F_2$, and a moving point $P$. The coordinates of $F_1$ and $F_2$ are $F_1(-1, 0)$ and $F_2(1, 0)$, respectively. The moving point $P$ satisfies $\frac{|PF_{1}|}{|PF_{2}|} = \frac{\sqrt{2}}{2}$. The trajectory of point $P$ is a curve $C$. The symmetrical curve of $C$ with respect to the line $y=x$ is curve $C''$. The line $y=x+m-3$ intersects curve $C''$ at points $A$ and $B$. Let $O$ be the origin. The area of the triangle $\triangle ABO$ is $\sqrt{7}$.  
(1) Find the equation of curve $C$.  
(2) Determine the value of $m$.

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