
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 the line \( l: x = my + q, m \neq 0 \) and the ellipse \(\Gamma: 3x^{2} + 4y^{2} = 12\) intersecting at two distinct points \( A \) and \( B \). Let \( A_1 \) be the point symmetric to \( A \) about the major axis of the ellipse, and \( F \) be the right focus of the ellipse. Find the necessary and sufficient condition for the points \( A_1, F, \) and \( B \) to be collinear.

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