
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
Niki and Kyle play a triangle game. Niki first draws  $\triangle ABC$  with area  $1$ , and Kyle picks a point  $X$  inside  $\triangle ABC$ . Niki then draws segments  $\overline{DG}$ ,  $\overline{EH}$ , and  $\overline{FI}$ , all through  $X$ , such that  $D$  and  $E$  are on  $\overline{BC}$ ,  $F$  and  $G$  are on  $\overline{AC}$ , and  $H$  and  $I$  are on  $\overline{AB}$ . The ten points must all be distinct. Finally, let  $S$  be the sum of the areas of triangles  $DEX$ ,  $FGX$ , and  $HIX$ . Kyle earns  $S$  points, and Niki earns  $1-S$  points. If both players play optimally to maximize the amount of points they get, who will win and by how much?

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