
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
Let  $ABC$  be a triangle.  Points  $D$ ,  $E$ , and  $F$  are respectively on the sides  $\overline{BC}$ ,  $\overline{CA}$ , and  $\overline{AB}$  of  $\triangle ABC$ .  Suppose that
\[
  \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x
\]
for some  $x$  with  $\frac{1}{2} < x < 1$ .  Segments  $\overline{AD}$ ,  $\overline{BE}$ , and  $\overline{CF}$  cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals.  The total area of the 4 triangles equals the total area of the 3 quadrilaterals.  Compute the value of  $x$ .

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