
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 $\overrightarrow{a}=3 \overrightarrow{e_{1}}-2 \overrightarrow{e_{2}}, \overrightarrow{b}=4 \overrightarrow{e_{1}}+ \overrightarrow{e_{2}}$, where $\overrightarrow{e_{1}}=(1,0), \overrightarrow{e_{2}}=(0,1)$, find:
$(1) \overrightarrow{a}\cdot \overrightarrow{b}$; $|\overrightarrow{a}+ \overrightarrow{b}|$;
$(2)$ The cosine value of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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