
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 $\overrightarrow{a} =(\cos \theta ,\sin \theta )$ and $\overrightarrow{b} =(2,-1)$.

$(1)$ If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\dfrac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }$;

$(2)$ If $|\overrightarrow{a} -\overrightarrow{b} |=2$ and $\theta \in (0,\dfrac{\pi }{2})$, find the value of $\sin (\theta +\dfrac{\pi }{4})$.

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