
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 non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $\overrightarrow{a}•\overrightarrow{b}=3$, $|{\overrightarrow{b}}|=2$, and $\overrightarrow{a}•({\overrightarrow{a}-3\overrightarrow{b}})=0$.  
$(Ⅰ)$ Find $|{\overrightarrow{a}}|$;  
$(Ⅱ)$ Find the angle $\theta$ between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$;  
$(Ⅲ)$ Find the value of $|{\overrightarrow{a}+\overrightarrow{b}}|$.

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