
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 denoted as $a$, $b$, and $c$, respectively, and it is given that $a^2 - (b-c)^2 = bc$, $\cos A \cos B = \frac{\sin A + \cos C}{2}$.

(1) Find the magnitudes of angles $A$ and $B$;  
(2) If $f(x) = \sin(2x+C)$, and the graph of $y=f(x)$ is translated to the right by $\frac{\pi}{12}$ units and then upwards by 2 units to obtain the graph of $y=g(x)$, find the expression for $g(x)$ and its interval of monotonic decrease.

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