
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
The function $f(x)=A\sin(\omega x+\varphi)$ $(A>0, 0<\omega<16, 0<\varphi<\frac{\pi}{2})$ has a maximum value of $\sqrt{2}$ on $R$, and $f(0)=1$.
(1) If the point $(\frac{\pi}{8}, \sqrt{2})$ is on the graph of $f(x)$, find the center of symmetry of the graph of the function $f(x)$;
(2) Shift the graph of the function $y=f(x)$ to the right by $\frac{\pi}{4\omega}$ units, then keep the ordinate unchanged, and shrink the abscissa to half of the original, obtaining the graph of the function $y=g(x)$. If $y=g(x)$ is an increasing function on $[0, \frac{\pi}{8}]$, find the maximum value of $\omega$.

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