
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
In the Cartesian coordinate plane, points $A(2,0)$, $B(0,2)$, and $C(\cos \alpha,\sin \alpha)$ are given.
(1) If $| \overrightarrow{AC}|=| \overrightarrow{BC}|$ and $\alpha \in (0, \pi)$, find the value of angle $\alpha$;
(2) If $\overrightarrow{AC} \cdot \overrightarrow{BC} = \dfrac{1}{3}$, find the value of $\dfrac{2\sin^{2}\alpha + \sin 2\alpha}{1 + \tan \alpha}$.

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