
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 $f(x) = \log_{4}(4^{x} + 1) + kx$ $(k \in \mathbb{R})$ is an even function.

$(1)$ Find the value of $k$;

$(2)$ If the graph of the function $y = f(x)$ has no intersection points with the line $y = \frac{1}{2}x + a$, find the range of values for $a$;

$(3)$ If the function $h(x) = 4^{f(x) + \frac{1}{2}x} + m \cdot 2^{x} - 1$, $x \in [0, \log_{2}3]$, is there a real number $m$ such that the minimum value of $h(x)$ is $0$? If it exists, find the value of $m$; if not, please explain why.

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