
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 an equilateral triangle $ABC$ with side length $2$, $\overrightarrow{BC} = 2\overrightarrow{BD}$, $\overrightarrow{CA} = 3\overrightarrow{CE}$. Let $\overrightarrow{AB} = \overrightarrow{a}$, $\overrightarrow{AC} = \overrightarrow{b}$.
(I) Express $\overrightarrow{AD}$ and $\overrightarrow{BE}$ in terms of $\overrightarrow{a}$ and $\overrightarrow{b}$.
(II) Find the value of $\overrightarrow{AD} \cdot \overrightarrow{BE}$.

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