
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 $|\overrightarrow{a}|=1, |\overrightarrow{b}|=2$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi}{3}$. Let $\overrightarrow{m}=3\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{n}=t\overrightarrow{a}+2\overrightarrow{b}$.  
$(1)$ If $\overrightarrow{m} \perp \overrightarrow{n}$, find the value of the real number $t$;  
$(2)$ If $t=2$, find the angle between $\overrightarrow{m}$ and $\overrightarrow{n}$.

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