
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 points $A(1,0)$, $B(0,1)$, and $C(2\sin \theta, \cos \theta)$.

$(1)$ If $|\overrightarrow{AC}|=|\overrightarrow{BC}|$, find the value of $\dfrac{\sin \theta+2\cos \theta}{\sin \theta-\cos \theta}$;  

$(2)$ If $(\overrightarrow{OA} +2 \overrightarrow{OB}) \cdot \overrightarrow{OC} =1$, where $O$ is the origin of coordinates, find the value of $\sin \theta \cdot \cos \theta$.

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