
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 $\overrightarrow{a}=({\cos\alpha,\sin\alpha})$ and $\overrightarrow{b}=({-\sin\alpha,\cos\alpha})$, let $\overrightarrow{m}=\sqrt{3}\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{n}=\overrightarrow{a}+\sqrt{3}\overrightarrow{b}$.  
$(1)$ Find the value of $|{\overrightarrow{a}+\overrightarrow{b}}|$.  
$(2)$ Find the angle between $\overrightarrow{m}$ and $\overrightarrow{n}$.

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