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

$(1)$ Find the standard equation of the ellipse;

$(2)$ Through point $F_{2}$, draw a line $l$ that does not coincide with the x-axis. Suppose $l$ 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 $\vec{F_{1}A} \cdot \vec{F_{1}B} = \lambda$ and $\lambda \in \left[ \frac{2}{3},1\right]$, find the range of the area $S$ of $\triangle F_{1}CD$.

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