
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 the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$. Here, $BC$ is extended beyond $C$ to a point $F$ such that $\angle BCF = 2\theta$. Determine a necessary and sufficient condition for the equality of the area of triangle $ABF$ and the remaining area of the circle excluding the sector $\overset{\frown}{BDF}$ when $0 < \theta < \frac{\pi}{2}$.

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