
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
In a triangle $\triangle ABC$ with points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$, respectively, where $\overline{CD} = \frac{1}{4}$ of $BC$, $\overline{AE} = \frac{1}{2}$ of $CA$, and $\overline{BF} = \frac{1}{3}$ of $AB$. Points $N_1$, $N_2$, and $N_3$ are the intersection points of $\overline{AD}$ and $\overline{CF}$, $\overline{BE}$ and $\overline{AD}$, $\overline{CF}$ and $\overline{BE}$ respectively. Calculate the area of $\triangle N_1N_2N_3$ in terms of the area of $\triangle ABC$, denoted as $K$.

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