
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 $f(x)=\sqrt{2}\sin(\omega x+\varphi)$, where $\omega\gt 0$ and $0\lt \varphi\lt \pi$. The distance between two adjacent highest points on the graph of this function is $4\pi$, and $f(x)$ is an even function.  
$(1)$ Find the values of $\omega$ and $\varphi$;  
$(2)$ In triangle $ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, if $(2a-c)\cos B=b\cos C$, find the range of $f^{2}(A)+f^{2}(C)$.

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