
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
Given a right triangle \( ABC \) with \(\angle A = 60^\circ\) and hypotenuse \( AB = 2 + 2\sqrt{3} \), a line \( p \) is drawn through vertex \( B \) parallel to \( AC \). Points \( D \) and \( E \) are placed on line \( p \) such that \( AB = BD \) and \( BC = BE \). Let \( F \) be the intersection point of lines \( AD \) and \( CE \). Find the possible value of the perimeter of triangle \( DEF \).

Options:

\[
\begin{gathered}
2 \sqrt{2} + \sqrt{6} + 9 + 5 \sqrt{3} \\
\sqrt{3} + 1 + \sqrt{2} \\
9 + 5 \sqrt{3} + 2 \sqrt{6} + 3 \sqrt{2}
\end{gathered}
\]

\[
1 + \sqrt{3} + \sqrt{6}
\]

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