Let function =y

If b is not equal to 0

y=(a+sinx)/(3+cosx) +bx

Obviously, the left part has a period and the right part is a monotone function.

and

When x→ positive infinity, y→ positive infinity.

Contradictions, then b is 0.

So the function is

y=(a+sinx)/(3+cosx)

sinx-ycosx=3y-a

Boundedness based on auxiliary angle formula and trigonometric function

(3y-a)^2<； =y^2+ 1

that is

8y^2-6ay+a^2- 1<； =0

According to the meaning of the question

The sum of the two roots of this equation is 6.

6a/8=6

The solution is a=8.