
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
(1) Given $\tan \theta=3$, find the value of $\dfrac {\sin \theta+\cos \theta}{2\sin \theta+\cos \theta}$;
(2) Given $0 < \beta < \dfrac {\pi}{2} < \alpha < \pi$, and $\cos (\alpha- \dfrac {\beta}{2})=- \dfrac {1}{9}$, $\sin ( \dfrac {\alpha}{2}-\beta)= \dfrac {2}{3}$, find the value of $\cos \dfrac {\alpha+\beta}{2}$.

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