
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]


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Given vectors $\overrightarrow{a}=({\sin x,0})$, $\overrightarrow{b}=({\cos x,\sin x})$, and the function $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}+{\overrightarrow{a}^2}$.  
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.  
$(2)$ If $A$, $B$, $C$ are the three interior angles of $\triangle ABC$ and $f({\frac{A}{2}})=1$, find the range of values for $f(B)$.

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