
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 vectors $\vec{a}$ and $\vec{b}$ with magnitudes $|\vec{a}|=2$ and $|\vec{b}|=\sqrt{3}$, respectively, and the equation $( \vec{a}+2\vec{b}) \cdot ( \vec{b}-3\vec{a})=9$:

(1) Find the dot product $\vec{a} \cdot \vec{b}$.

(2) In triangle $ABC$, with $\vec{AB}=\vec{a}$ and $\vec{AC}=\vec{b}$, find the length of side $BC$ and the projection of $\vec{AB}$ onto $\vec{AC}$.

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