
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 an ellipse $C$ with foci $F_1(-2\sqrt{2},0)$, $F_2(2\sqrt{2},0)$, and a moving point $P$ on it satisfying $|PF_1|+|PF_2|=4\sqrt{3}$. Let $O$ be the origin, and the bottom vertex of the ellipse $C$ be $R$.

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

$(2)$ Suppose a line $l_2$ passing through the point $(0,1)$ with a slope of $k$ intersects the ellipse $C$ at points $M$ and $N$, investigate whether $\vec{RM} \cdot \vec{RN}$ is always a constant value regardless of the value of $k$. If yes, find the constant value; if no, explain why.

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