
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $O$ is the origin, and point $P(-1,\frac{\sqrt{2}}{2})$ lies on the ellipse, with the eccentricity of the ellipse being $\frac{\sqrt{2}}{2}$.

(1) Find the standard equation of the ellipse;

(2) Let $\bigcirc O$ be the circle with diameter $F_1F_2$. A line $l: y=kx+m$ is tangent to $\bigcirc O$ 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}
user