
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 functions $f(x)=\sin \omega x$, $g(x)=\sqrt{3}\cos \omega x$ $(\omega > 0)$, and $h(x)=3\cos^2 \omega x + \sqrt{3}\sin \omega x \cos \omega x$. Choose one of the following three conditions about the graphs of functions $f(x)$ and $g(x$ as known, so that the function $h(x)$ is uniquely determined, and solve the following problems.<br/>$(1)$ Find the analytical expression of the function $h(x)$;<br/>$(2)$ If for $\forall x\in \mathbb{R}$, there exists a unique $a\in [0,m]$, such that $h(a-x)=h(a+x)$, find the range of $m$.<br/>Condition 1: The graphs of the two functions have exactly two intersection points in the interval $[0,2\pi]$;<br/>Condition 2: The horizontal distance between adjacent intersection points of the two graphs is $\pi$;<br/>Condition 3: The minimum horizontal distance between the highest points of the two graphs is $\frac{\pi}{2}$.<br/>Note: If multiple sets of conditions are selected for separate answers, the first answer will be scored.

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