
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 the foci of the ellipse $(C)$ are $F_1(-2\sqrt{2},0)$ and $F_2(2\sqrt{2},0)$, and a moving point $P$ on the ellipse satisfies $|PF_1|+|PF_2|=4\sqrt{3}$. Point $O$ is the coordinate origin, and the lower vertex of the ellipse $(C)$ is $R$.

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

(2) Let line $(l_2)$ pass through point $(0,1)$ with a slope of $k$, intersecting the ellipse $(C)$ at points $M$ and $N$. Investigate whether $\overrightarrow{RM} \cdot \overrightarrow{RN}$ is always constant, regardless of the value of $k$. If it is constant, find the value.

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