
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 that one focus of the ellipse $C_{1}： \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a > b > 0)$ coincides with the focus of the parabola $C_{2}：y^{2}=2px (p > 0)$, and the distance from point F to the line $x - y + 1 = 0$ is $\sqrt {2}$, and the length of the common chord of $C_{1}$ and $C_{2}$ is $2 \sqrt {6}$.
(1) Find the coordinates of F;
(2) Find the equation of the ellipse $C_{1}$.

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