
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 that the terminal side of angle $\theta$ lies on the ray $y=2x(x\geqslant 0)$.  
$(1)$ Find the value of $\tan \theta$;  
$(2)$ Find the value of $\dfrac{2\cos \theta +3\sin \theta }{\cos \theta -3\sin \theta }+\sin \theta \cos \theta$.

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