
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 a direction \(\vec{l}\) in a plane \(\alpha\), let \( F \) be a convex set within the plane \(\alpha\) with an area of \( S(F) \). Among all the triangles inscribed in \( F \) with one side parallel to \(\vec{l}\), let \(\triangle\) be the one with the largest area, and let its area be denoted as \( S(\Delta) \). Find the maximum positive real number \( c \) such that for any convex shape \( F \) in the plane \(\alpha\), it holds that \( S(\Delta) \geqslant c \cdot S(F) \).

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