
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 ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $e= \frac{1}{2}$, its right focus is $F(c,0)$. The two real roots of the equation $ax^{2}+bx-c=0$ are $x_{1}$ and $x_{2}$, respectively. Determine the position of point $P(x_{1},x_{2})$ relative to the circle $x^{2}+y^{2}=2$.

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