
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 equation $\arctan \dfrac {x}{2}+\arctan (2-x)=a$;  
$(1)$ If $a= \dfrac {\pi}{4}$, find the value of $\arccos \dfrac {x}{2}$;  
$(2)$ If the equation has real solutions, find the range of values for the real number $a$;  
$(3)$ If the equation has two distinct solutions $\alpha$ and $\beta$ in the interval $[5,15]$, find the maximum value of $\alpha+\beta$.

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