
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 $\alpha$ lies in the second quadrant, and the ordinate of the intersection point with the unit circle is $\frac{2\sqrt{5}}{5}$, when the terminal side of angle $\alpha$ is rotated counterclockwise by $\frac{π}{2}$, it coincides with the terminal side of angle $\beta$.  
$(1)$ Find $cos(\alpha+\frac{2π}{3})$;  
$(2)$ Find the value of $\frac{sin\alpha-cos(\alpha-π)}{sin\beta+2sin(\frac{π}{2}-\beta)}$.

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