
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 triangle $ABC$, $AB$ is congruent to $AC$, and the measure of angle $ABC$ is $60^{\circ}$. Segment $BD$ bisects angle $ABC$ with point $D$ on side $AC$. Point $E$ is on side $BC$ such that segment $DE$ is parallel to side $AB$, and point $F$ is on side $AC$ such that segment $EF$ is parallel to segment $BD$ and $EF$ bisects $\angle DEC$. How many isosceles triangles are present?

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