
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 triangle \(ABC\) with an area of 1. The first player chooses a point \(X\) on side \(AB\), the second player chooses a point \(Y\) on side \(BC\), and then the first player chooses a point \(Z\) on side \(AC\). The first player's goal is to maximize the area of triangle \(XYZ\), and the second player's goal is to minimize it. What is the largest area that the first player can guarantee for themselves?

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