
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 that in $\triangle ABC$, $\angle BAC = \frac{2\pi}{3}$, $AB = 2$, $AC = 1$, $D$ is a point on side $BC$ such that $DC = 2BD$. Find the value of $\overrightarrow{AD} \cdot \overrightarrow{BC}$.

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