
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
Gizi and Tibi play the following game. Gizi marks a point $X$ on side $AB$ of triangle $ABC$, then Tibi selects a point $Y$ on side $BC$. Finally, Gizi picks a point $Z$ on $AC$. Gizi aims to maximize the area of triangle $XYZ$, while Tibi aims to minimize it. What will be the area of triangle $XYZ$ if both play optimally?

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