
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
Let $f(x)=\sin x+\cos x$ for $x\in \mathbb{R}$.  
$(1)$ Find the smallest positive period of the function $y={[f(x+\frac{π}{2})]}^{2}$.  
$(2)$ Shift the graph of the function $y=f(x)$ to the right by $\frac{π}{2}$ units to obtain the function $g(x)$. Find the maximum value of $g(x)$ on the interval $\left[0,\pi \right]$.

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