
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
(I) Given the ellipse equation $\frac{x^2}{4} + \frac{y^2}{3} = 1$ and point $P\left( 0,\sqrt{3} \right)$. Let points $A_{2}\left( \sqrt{3},\frac{\sqrt{3}}{2} \right)$ and $B_{2}\left( -\sqrt{3},-\frac{\sqrt{3}}{2} \right)$ be symmetric about the origin. Denote the slopes of lines $PA_{2}$ and $PB_{2}$ as $k_{PA_{2}}$ and $k_{PB_{2}}$, respectively. Calculate the value of $k_{PA_{2}} \cdot k_{PB_{2}}$.

(II) Based on the conclusion from part (I), explore the conjecture: Let $M$ and $N$ be two points on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, symmetric about the origin. Let $Q$ be any point on the ellipse such that the slopes of lines $QM$ and $QN$, denoted by $k_{QM}$ and $k_{QN}$, both exist. Conjecture the value of $k_{QM} \cdot k_{QN}$ and provide a proof.

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