
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 $f(x)=\log_{2}(2^{x}+1)+kx$ where $k\in\mathbb{R}$, and it satisfies $f(-1)=f(1)$.  
$(1)$ Find the value of the real number $k$.  
$(2)$ If the graph of the function $y=f(x)$ intersects the line $y=\frac{1}{2}x+a$ at only one point, find the range of values for $a$.  
$(3)$ For the function $h(x)=2^{f(x)+\frac{1}{2}x}+m\cdot 4^{x}-1$, where $x\in[0,\log_{2}3]$, does there exist a real number $m$ such that the minimum value of $h(x)$ is $0$? If it exists, find the value of $m$; if not, explain the reason.

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