
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
In \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, and it is given that \\(a^{2}-(b-c)^{2}=bc\\) and \\(\cos A\cos B=\dfrac{\sin A+\cos C}{2}\\).

\\((1)\\) Find the magnitudes of angles \\(A\\) and \\(B\\).

\\((2)\\) If \\(f(x)=\sin (2x+C)\\), and the graph of the function \\(y=f(x)\\) is translated to the right by \\(\dfrac{\pi}{12}\\) units and then upwards by \\(2\\) units to obtain the graph of the function \\(y=g(x)\\), find the expression for \\(g(x)\\) and its interval of monotonic decrease.

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