
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
Nóra and Kristóf play the following game: On a given triangle $ABC$, Nóra designates a point $N_{1}$ on side $AB$. Then Kristóf chooses a point $K$ on side $BC$. Finally, Nóra marks a point $N_{2}$ on side $CA$. Nóra aims to maximize the area of triangle $N_{1}KN_{2}$, while Kristóf aims to minimize it. What will be the area of triangle $N_{1}KN_{2}$ if both play optimally?

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