
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 its left and right foci located at $F_{1}(-c,0)$ and $F_{2}(c,0)$, respectively, and a point $A\left(1, \frac{\sqrt{2}}{2}\right)$ on the ellipse $C$.

(1) Find the standard equation of the ellipse $C$.

(2) Determine whether there exists a line $l$ with slope $2$ such that when $l$ intersects the ellipse $C$ at two distinct points $M$ and $N$, a point $P$ can be found on the line $y= \frac{5}{3}$ and a point $Q$ on the ellipse $C$ satisfying $\overrightarrow{PM}=\overrightarrow{NQ}$. If such a line exists, find its equation; if not, explain why.

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