
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
In the rectangular coordinate system $xOy$, $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{4} = 1$ with $a > 0$. The line $l$ passes through the right vertex $A$ of the hyperbola and the point $B(0, 2)$. If the sum of the distances from points $F_{1}$ and $F_{2}$ to the line $l$ is equal to the length of the segment $B F_{2}$, then the coordinates of point $A$ are $\qquad$

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