
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 points $A(-2,0)$ and $B(2,0)$, the slope of line $PA$ is $k_1$, and the slope of line $PB$ is $k_2$, with the product $k_1k_2=-\frac{3}{4}$.
$(1)$ Find the equation of the locus $C$ for point $P$.
$(2)$ Let $F_1(-1,0)$ and $F_2(1,0)$. Extend line segment $PF_1$ to meet the locus $C$ at another point $Q$. Let point $R$ be the midpoint of segment $PF_2$, and let $O$ be the origin. Let $S$ represent the sum of the areas of triangles $QF_1O$ and $PF_1R$. Find the maximum value of $S$.

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