
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 trapezoid $ABCD$, with $AB \parallel CD$, and $AB=AD=2$, $CD=3$.  
$(1)$ Express vector $\overrightarrow{BD}$ in terms of vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$.  
$(2)$ If $AD \perp AB$, find the cosine of the angle between vectors $\overrightarrow{AC}$ and $\overrightarrow{BD}$.

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