
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 that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $θ=\frac{{3π}}{4}$, and $|{\overrightarrow{a}}|=3$, $|{\overrightarrow{b}}|=2\sqrt{2}$.  
$(1)$ Find $\overrightarrow{a}⋅\overrightarrow{b}$ and $|{\overrightarrow{a}+\overrightarrow{b}}|$;  
$(2)$ Find the cosine value of the angle between $\overrightarrow{a}$ and $\overrightarrow{a}+\overrightarrow{b}$;  
$(3)$ If $\overrightarrow{b}=(2,2)$, find the projection vector of $\overrightarrow{a}$ onto $\overrightarrow{b}$.

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