
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 function $f(x)=\sin 2x+2\cos ^{2}x-1$, $x\in (0,\pi)$.
1. Find the interval where the function $y=f(x)$ is monotonically decreasing.
2. In triangle $ABC$, if $f(A)=f(B)$ and $A\neq B$, $AB=\sqrt {2}$, find the radius of the circumcircle of triangle $ABC$.

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