
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 two points in the plane, $A(0,-a)$ and $B(0,a)$ ($a > 0$), there is a moving point $P$ in the plane, and the slopes of line $PA$ and line $PB$ are $k_{1}$ and $k_{2}$, respectively. Let $k_{1} \cdot k_{2} = m$, where $m \neq 0$.
- (I) Find the equation of the trajectory of point $P$;
- (II) Given that point $N$ is on the circle $x^{2}+y^{2}=a^{2}$, and when $m \in (-1,0)$, the corresponding curve is $C$. Let $F_{1}$ and $F_{2}$ be the two foci of this curve. Is there a point $N$ such that the area of $\triangle F_{1}NF_{2}$ is $S= \sqrt {-m} \cdot a^{2}$?

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