
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 the following conditions:①$sin(π+α)=\frac{2\sqrt{5}}{5}$, ②$cos(2π-α)=\frac{\sqrt{5}}{5}$, ③$cos2α=-\frac{3}{5}$, choose one of these three known conditions and complete the statement below, then provide the solution.<br/>Question: Given that angle $\alpha$ is a fourth quadrant angle and satisfies _____.<br/>$(1)$ Find the value of $cos(\frac{π}{3}+α)$;<br/>$(2)$ If $tan(α+β)=\frac{1}{7}$, find the value of $\tan \beta$.

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