
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 $C_{1}： \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a＞b＞0)$, with left focus $F$, upper vertex $A$, and right vertex $B(2,0)$. If $\sqrt {7}| \overrightarrow {AF}|=2| \overrightarrow {AB}|$, and the vertex of the parabola $C_{2}$ is at the origin with focus $F$.

(1) Find the standard equation of $C_{1}$;
(2) Does there exist a line passing through $F$ that intersects $C_{1}$ and $C_{2}$ at points $P$, $Q$ and $M$, $N$ respectively, such that $S_{△OPQ}= \frac {1}{2}S_{△OMN}$? If it exists, find the equation of the line; if not, explain the reason.

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