
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 the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $O$ is the coordinate origin, and point $P(-1,\frac{\sqrt{2}}{2})$ is on the ellipse. The line segment $PF_{2}$ intersects the $y$-axis at point $M$, such that $\overrightarrow{PM}=\overrightarrow{F_{2}M}=\overrightarrow{o}$.

(1) Find the standard equation of the ellipse.

(2) $\odot O$ is a circle with $F_{1}F_{2}$ as its diameter. A line $l$: $y=kx+m$ is tangent to $\odot O$ and intersects the ellipse at two distinct points $A$ and $B$. When $\overrightarrow{OA} \cdot \overrightarrow{OB} = \lambda$ and $\frac{2}{3} \leqslant \lambda \leqslant \frac{3}{4}$, find the range of values for the area $S$ of $\triangle AOB$.

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