
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 $f(x)=a\sin \left( 2\omega x+\frac{\pi}{6} \right)+\frac{a}{2}+b$ $(x \in \mathbb{R}, a > 0, \omega > 0)$ has a minimum positive period of $\pi$, and the maximum value of function $f(x)$ is $\frac{7}{4}$, while the minimum value is $\frac{3}{4}$.

$(1)$ Find the values of $\omega$, $a$, and $b$;

$(2)$ Identify the intervals where $f(x)$ is monotonically increasing.

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