
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 $f(x)=2\sin^2\left(\frac{\pi}{4}+x\right)-\sqrt{3}\cos 2x-1$, where $x\in\mathbb{R}$,

$(1)$ The graph of the function $h(x)=f(x+t)$ is symmetric about the point $\left(-\frac{\pi}{6},0\right)$, and $t\in(0,\pi)$. Find the value of $t$;

$(2)$ For $x\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, it always holds that $|f(x)-m| < 3$. Find the range of the real number $m$.

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