
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
On sides $A B$ and $A C$ of triangle $A B C$, points $E$ and $F$ are taken, respectively. Segments $B F$ and $C E$ intersect at point $D$. Find the area of triangle $A E F$ if the areas of triangles $B C D$, $B D E$, and $C D F$ are $1$, $\frac{1}{2}$, and $\frac{1}{6}$, respectively.

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