
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 $F_1$ and $F_2$ are the left and right foci of an ellipse, $O$ is the coordinate origin, and point $P\left(-1, \frac{\sqrt{2}}{2}\right)$ is on the ellipse. The line segment $PF_2$ intersects the $y$-axis at point $M$, which satisfies $\overrightarrow{PM}+\overrightarrow{F_2M}=0$.

(I) Find the standard equation of the ellipse.

A circle $O$ has $F_1F_2$ as its diameter. A line $l:y=kx+m$ is tangent to the circle and intersects the ellipse at two distinct points $A$ and $B$. When $\overrightarrow{OA}\cdot\overrightarrow{OB}=\lambda$ and $\frac{2}{3}\leq\lambda\leq\frac{3}{4}$, find the range of values for the area $S$ of $\triangle OAB$.

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