
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 a focal distance of 2 and eccentricity $e$ of $$\frac{1}{2}$$:
1. Determine the standard equation of the ellipse $C$.
2. Given the point $$P\left(\frac{1}{2}, 1\right)$$, consider the circle $$O: x^2 + y^2 = \frac{1}{2}$$, and draw tangents at points $M$ and $N$. Line $MN$ intersects the $x$-axis at point $F$. A line $l$ passing through point $F$ intersects the ellipse $C$ at points $A$ and $B$. If $G$ is the reflection of $F$ across the $y$-axis, find the maximum area of triangle $△ABG$.

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