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

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