
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 $2\cos \theta +\sin \theta =0$, and $\theta \in (0,\pi)$.

        (I) Find the values of $\tan \theta$, $\sin \theta$, and $\cos \theta$ respectively;

        (II) If $\sin (\theta -\varphi) =\frac{\sqrt{10}}{10}$, $\frac{\pi }{2} < \varphi < \pi$, find the value of $\cos \varphi$.

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