
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
Let $x$, $y\in \mathbb{R}$, vectors $\overrightarrow{a}=(2,x)$, $\overrightarrow{b}=(y,1)$, $\overrightarrow{c}=(3,-3)$, and $\overrightarrow{a}⊥\overrightarrow{b}$, $\overrightarrow{b}∥\overrightarrow{c}$.  
$(1)$ Find $|\overrightarrow{a}+\overrightarrow{b}|$;  
$(2)$ Find the cosine value of the angle between vectors $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}+2\overrightarrow{b}+\overrightarrow{c}$.

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