
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 that the domain of the function $y=f(x)$ is $D$, and the derivative of the function is $y=f'(x)$, if for any $x\in D$, we have $f(x) < f'(x)$, then the function $y=f(x)$ is called an "$M$-type function" on $D$.  
$(1)$ Determine whether $f(x)=\sin x$ is an "$M$-type function" on its domain, and explain the reason.  
$(2)$ If the function $g(x)=ax+a-1$, $x\in (0,\pi)$ is an "$M$-type function" on its domain, find the range of the real number $a$.  
$(3)$ Given that the function $h(x)=\sin x+ax+a-2$, $x\in[0,\frac{\pi}{2}]$ is an "$M$-type function" on its domain, find the maximum integer value of the real number $a$.

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