
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 \( F_{1} \) and \( F_{2} \) are the left and right foci of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\), and \( A \) is a moving point on the ellipse. Circle \( C \) is tangent to the extended line of \( F_{1}A \), the extended line of \( F_{1}F_{2} \), and segment \( AF_{2} \). If \( M(t, 0) \) is a point of tangency, then \( t=\_\_\_\_\_ \).

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