
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
In the rectangular coordinate system $(xOy)$, the line $x+y-2=0$ passing through the right focus $F$ of the ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a > b > 0)$ intersects the ellipse $C$ at points $A$ and $B$. $P$ is the midpoint of $AB$, and the slope of $OP$ is $\frac{{1}}{{3}}$.

1. Find the standard equation of the ellipse $C$.
2. Suppose a line $l$ (not perpendicular to the coordinate axes) passes through point $F$ and intersects the ellipse $C$ at points $D$ and $E$. Is there a fixed point $M$ on the $x$-axis such that $\overrightarrow{MD}\cdot \overrightarrow{ME}$ is a constant value? If it exists, find the coordinates of point $M$; if not, explain the reason.

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