
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 $f(x)=\sin(\omega x+\varphi)-1$ where $\omega > 0$ and $0 < \varphi < \pi$, the distance between two adjacent symmetry axes of the graph is $\frac{\pi}{2}$. If we first shift each point on the graph of $f(x)$ to the left by $\frac{\pi}{12}$ units and then up by $1$ unit, the resulting function $g(x)$ is an even function.  
$(1)$ Find the analytical expression of $f(x)$.  
$(2)$ If for any $x \in [0,\frac{\pi}{3}]$, $\left[f(x)\right]^2-(2+m)f(x)+2+m \leq 0$ always holds, find the range of real numbers $m$.  
$(3)$ If the graph of the function $h(x)=2f(x)+3$ has at least $30$ zeros in the interval $\left[a,b\right]$ where $a$ and $b$ are real numbers and $a < b$, find the minimum value of $b-a$.

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