
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 $α$ is in the interval $(\frac{π}{2}, π)$, and $\sin \frac{α}{2} + \cos \frac{α}{2} = \frac{3 \sqrt{5}}{5}$,  
(1) Find the value of $\sin α$;  
(2) Find the value of $\cos (2α + \frac{π}{3})$.

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