You are a helpful assistant. When the user asks a question, your response must include two parts: first, the reasoning process enclosed in <think>...</think> tags, then the final answer enclosed in <answer>...</answer> tags.Please provide a clear, concise response within <answer> </answer> tags that directly addresses the question.
system
<image>Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ$, $\angle ABC = 60^\circ$, and $\angle ACB = 36^\circ$. Let $D$, $E$, and $F$ be the midpoints of sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD}$, $\overline{AE}$, and $\overline{AF}$ at points $G$, $H$, and $J$, respectively. The points $G$, $D$, $E$, $H$, $J$, and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\wideparen{DE} + 2 \cdot \wideparen{HJ} + 3 \cdot \wideparen{FG}$, where the arcs are measured in degrees.

\begin{tikzpicture}[scale=1.2]
    \coordinate (B) at (0,0);
    \coordinate (C) at (6,0);
    \coordinate (A) at (1.78,3.07);
    
    \coordinate (D) at ($(B)!0.5!(C)$);
    \coordinate (E) at ($(A)!0.5!(C)$);
    \coordinate (F) at ($(A)!0.5!(B)$);
    
    \draw (A) -- (B) -- (C) -- cycle;

    \fill (A) circle (2pt) node[above] {$A$};
    \fill (B) circle (2pt) node[below] {$B$};
    \fill (C) circle (2pt) node[below] {$C$};

    \fill (D) circle (2pt) node[below] {$D$};
    \fill (E) circle (2pt) node[right] {$E$};
    \fill (F) circle (2pt) node[left] {$F$};

    \draw (D) -- (E) -- (F) -- cycle;

    \coordinate (O) at (2.39,1.38);
    
    \draw (O) circle (1.51);

    \coordinate (G) at (1.72, 0.03);
    \coordinate (J) at (1.5, 2.6);
    \coordinate (H) at (2.08, 2.85);

    \fill (G) circle (2pt) node[below] {$G$};
    \fill (H) circle (2pt) node[above] {$H$};
    \fill (J) circle (2pt) node[above left] {$J$};
\end{tikzpicture}
 You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
user
336
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ$, $\angle ABC = 60^\circ$, and $\angle ACB = 36^\circ$. Let $D$, $E$, and $F$ be the midpoints of sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD}$, $\overline{AE}$, and $\overline{AF}$ at points $G$, $H$, and $J$, respectively. The points $G$, $D$, $E$, $H$, $J$, and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\wideparen{DE} + 2 \cdot \wideparen{HJ} + 3 \cdot \wideparen{FG}$, where the arcs are measured in degrees.

\begin{tikzpicture}[scale=1.2]
    \coordinate (B) at (0,0);
    \coordinate (C) at (6,0);
    \coordinate (A) at (1.78,3.07);
    
    \coordinate (D) at ($(B)!0.5!(C)$);
    \coordinate (E) at ($(A)!0.5!(C)$);
    \coordinate (F) at ($(A)!0.5!(B)$);
    
    \draw (A) -- (B) -- (C) -- cycle;

    \fill (A) circle (2pt) node[above] {$A$};
    \fill (B) circle (2pt) node[below] {$B$};
    \fill (C) circle (2pt) node[below] {$C$};

    \fill (D) circle (2pt) node[below] {$D$};
    \fill (E) circle (2pt) node[right] {$E$};
    \fill (F) circle (2pt) node[left] {$F$};

    \draw (D) -- (E) -- (F) -- cycle;

    \coordinate (O) at (2.39,1.38);
    
    \draw (O) circle (1.51);

    \coordinate (G) at (1.72, 0.03);
    \coordinate (J) at (1.5, 2.6);
    \coordinate (H) at (2.08, 2.85);

    \fill (G) circle (2pt) node[below] {$G$};
    \fill (H) circle (2pt) node[above] {$H$};
    \fill (J) circle (2pt) node[above left] {$J$};
\end{tikzpicture}
 You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
val