
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 $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ satisfy: $|\overrightarrow{a}|=2, |\overrightarrow{b}|=4, \overrightarrow{c}= \overrightarrow{a}- \overrightarrow{b}$, and $\overrightarrow{c} \perp \overrightarrow{a}$  
$(1)$ Find the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$;  
$(2)$ Find $\overrightarrow{a} \cdot (\overrightarrow{a}+3\overrightarrow{b})$ and $|3\overrightarrow{a}+\overrightarrow{b}|$.

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