
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 $f(x) = A\sin(\omega x + \varphi)$ ($A>0$, $\omega>0$, $|\varphi|< \frac{\pi}{2}$, $x\in\mathbb{R}$), the maximum value is 10, and the graph of $f(x)$ passes through the point $(0,5)$, with the distance between adjacent axes of symmetry being $\frac{\pi}{2}$.
(Ⅰ) Find the expression for $f(x)$.
(Ⅱ) After shifting the graph of $f(x)$ to the right by $\frac{\pi}{6}$ units, we obtain the graph of $g(x)$. Determine the intervals of increase for $g(x)$.

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