
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 circle with center $C$, and radius $r$, consider a point $A$ on the circle such that $AB$ is tangent to the circle at $A$. Let $D$ and $E$ be points on the circle such that $\angle BCD = \phi$ and $\angle ACE = 2\phi$, where $\phi$ is measured in radians and $0 < \phi < \frac{\pi}{4}$. Determine the necessary and sufficient condition for the shaded areas formed by sectors $BCD$ and $ACE$ and triangles $BCA$ and $ECA$ to be equal. Assume $BCD$ and $ACE$ are line segments connecting the respective points.

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