
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 the ellipse $W$: $\frac{x^{2}}{4}+y^{2}=1$, a line $l_{1}$ passing through the origin $O$ intersects the ellipse $W$ at points $A$ and $B$. $P$ is a moving point on the ellipse distinct from $A$ and $B$. The lines $PA$ and $PB$ have slopes $k_{1}$ and $k_{2}$ ($k_{1},k_{2} \neq 0$), respectively. Through $O$, draw lines $l_{2}$ and $l_{3}$ parallel to $PA$ and $PB$, which intersect the ellipse $W$ at points $C$, $D$ and $E$, $F$, respectively.

(I) If $A$ and $B$ are the left and right vertices of the ellipse $W$, does a point $P$ exist such that $\angle APB = 90^{\circ}$? Explain your reasoning.

(II) Find the value of $k_{1} \cdot k_{2}$.

(III) Find the value of $|CD|^{2}+|EF|^{2}$.

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