
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 $\triangle DEF$ be an isosceles triangle with $DE = DF$. Three circles are defined as follows: the circle $\Omega$ with its center at the centroid of $\triangle DEF$, and two circles $\Omega_1$ and $\Omega_2$, where $\Omega_1$ is tangent to $\overline{EF}$ and externally tangent to the other sides extended, while $\Omega_2$ is tangent to $\overline{DE}$ and $\overline{DF}$ while internally tangent to $\Omega$. Determine the smallest integer value for the perimeter of $\triangle DEF$ if $EF$ is the smallest integer side, and $\Omega_1$ and $\Omega_2$ intersect only once.

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