
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 $1$, let $\overrightarrow{BC}=2\overrightarrow{BD}$ and $\overrightarrow{CA}=3\overrightarrow{CE}$,

$(1)$ Find the projection of $\overrightarrow{AB}$ in the direction of $\overrightarrow{DA}$.

$(2)$ Find the value of $\overrightarrow{AD} \cdot \overrightarrow{BE}$.

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