
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 an ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with foci $F_{1}$, $F_{2}$, and a point $P$ on the ellipse $C$. If $PF_{1} \perp PF_{2}$, $|F_{1}F_{2}|=2 \sqrt {3}$, and the area of $\triangle PF_{1}F_{2}$ is $1$.
(1) Find the equation of the ellipse $C$;
(2) If there always exist two points $A$ and $B$ on the ellipse $C$ that are symmetric about the line $y=x+m$, find the range of values for the real number $m$.

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