
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 ellipse \( C: \frac{x^{2}}{4}+y^{2}=1 \) with the upper vertex \( M \) and the lower vertex \( N \), let \( T(t, 2) \) (where \( t \neq 0 \)) be a point on the line \( y=2 \). The lines \( TM \) and \( TN \) pass through \( T \) and intersect the ellipse \( C \) at points \( E \) and \( F \), respectively. If the area of \( \triangle TMN \) is \( k \) times the area of \( \triangle TEF \), what value of \( t \) maximizes \( k \)?

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