
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 the left and right focal points at $F_1(-c, 0)$ and $F_2(c, 0)$, respectively, given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and let it pass through the point $(\sqrt{3}, \frac{\sqrt{3}}{2})$. Furthermore, the symmetrical image of ellipse $C$ about the line $x = c$ passes through the origin. 
(I) Determine the eccentricity and standard equation of ellipse $C$. 
(II) Circle $P_1: \left(x + \frac{4\sqrt{3}}{7}\right)^2 + \left(y - \frac{3\sqrt{3}}{7}\right)^2 = r^2$ ($r > 0$) intersects ellipse $C$ at points $A$ and $B$. Let $R$ be any point on segment $AB$. The line $F_1R$ intersects ellipse $C$ at points $P$ and $Q$. If $AB$ is the diameter of circle $P_1$ and the slope of $F_1R$ is greater than 1, determine the range of $|PF_1|\cdot|QF_1|$.

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