
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
The side of the base $ABC$ of the pyramid $TABC$ is 4, and the lateral edge $TA$ is perpendicular to the base plane. Find the area of the cross-section of the pyramid with the plane passing through the midpoints of edges $AC$ and $BT$ parallel to the median $BD$ of face $BCT$, given that the distance from the vertex $T$ to this plane is $\frac{1}{2}$.

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