
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 $\overrightarrow{PM} + \overrightarrow{F_2M} = \overrightarrow{0}$.

(1) Find the standard equation of the ellipse;

(2) Circle $⊙O$ has diameter $F_1F_2$. A line $l: y = kx + m$ is tangent to the circle $⊙O$ and intersects the ellipse at two distinct points $A$ and $B$. When $\overrightarrow{OA} \cdot \overrightarrow{OB} = λ$ and $\frac{2}{3} \leqslant λ \leqslant \frac{3}{4}$, find the range of values for the area $S$ of $ΔAOB$.

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