
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 $$C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$$ with left and right foci $F_1$ and $F_2$ respectively, and an eccentricity of $$\frac{1}{2}$$. $P$ is a moving point on the ellipse such that the maximum area of triangle $PF_1F_2$ is $$\sqrt{3}$$.
(I) Find the equation of the ellipse $C$;
(II) Given that the slope of line $PF_2$ exists and intersects ellipse $C$ at another point $Q$, determine if there exists a point $T(0, t)$ such that $|TP| = |TQ|$. If it exists, find the range of $t$; if not, explain the reason.

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