
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 $\triangle ABC$, let $E$ be a point on side $BC$ such that $BE = 2EC$. Point $D$ lies on side $AC$ such that $AD = 2DC$. If the length of $AC$ is $2$, and $\measuredangle BAC = 45^\circ$, $\measuredangle ABC = 75^\circ$, and $\measuredangle ACB = 60^\circ$, calculate the sum of the area of $\triangle ABC$ and twice the area of $\triangle CDE$.
A) $\frac{2\sqrt{6} + 2\sqrt{2}}{3}$
B) $\frac{\sqrt{6} + \sqrt{2}}{3}$
C) $\frac{3\sqrt{6} + 3\sqrt{2}}{9}$
D) $\frac{\sqrt{6} + \sqrt{2}}{6}$
E) $\frac{3\sqrt{6} + 3\sqrt{2}}{6}$

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