
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 $\alpha$ intersects with the circle $x^{2}+y^{2}=3$ at point $P(m, \sqrt {2})$ in the first quadrant, find:
$(1)$ the value of $\tan \alpha$;
$(2)$ the value of $\dfrac {2\cos ^{2} \dfrac {\alpha}{2}-\sin \alpha-1}{ \sqrt {2}\sin ( \dfrac {\pi}{4}+\alpha)}$.

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