
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
Sound is the energy that can cause hearing due to the vibration of objects, and each sound is composed of pure tones. The mathematical model of a pure tone is $y=A\sin \omega t$. Generally, the musical sounds we hear are not just a single tone, but a composite sound composed of multiple waves. The combination of different vibrations determines the timbre of the sound, by which people distinguish different sounds. It is known that the function of a certain sound is $f(x)=asin^{2}x+\sqrt{3}asinxcosx-\frac{3}{2}a+b$ (where $a>0$, $b\in R$), and the amplitude of the function $f(x)$ is $4$.  
$(1)$ When $x\in[0,\frac{\pi}{2}]$, the maximum value of the function $f(x)$ is $1$. Find the values of the real numbers $a$ and $b$.  
$(2)$ Under the condition in $(1)$, find the axis of symmetry of the graph of the function $f(x)$ and the interval of monotonic increase on $(0,\pi)$.

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