
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and point $P(-1, \frac{\sqrt{2}}{2})$ is on the ellipse. The line segment $PF_2$ intersects the y-axis at point $M$ satisfying $\overrightarrow{PM} = \overrightarrow{MF_2}$.

(Ⅰ) Find the standard equation of the ellipse;

(Ⅱ) Draw a line $l$ through $F_2$ that does not coincide with the x-axis. Line $l$ intersects the circle $x^2 + y^2 = a^2 + b^2$ at points $A$ and $B$, and intersects the ellipse at points $C$ and $D$. When $\overrightarrow{F_1A} \cdot \overrightarrow{F_1B} = \lambda$, and $\lambda \in [\frac{2}{3}, 1]$, find the range of the area $S$ of $\triangle F_1CD$.

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