
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 center of the ellipse $C$ is at the origin of the coordinate, the focus is on the $x$-axis, the eccentricity $e= \frac {1}{2}$, and the ellipse $C$ passes through the point $P(2,3)$. A line passing through the left focus $F_{1}$ of the ellipse $C$ and not perpendicular to the coordinate axis intersects the ellipse $C$ at points $A$ and $B$.
(1) Find the equation of the ellipse $C$;
(2) Let the perpendicular bisector of segment $AB$ intersect the $x$-axis at point $G$, find the range of values for the area $S$ of $\triangle PF_{1}G$.

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