
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 $f(x) = \log_{4}(4^{x}+1) + kx$, where $k \in \mathbb{R}$, the graph of $f(x)$ is symmetric about the y-axis.  
(1) Find the value of the real number $k$;  
(2) If the equation $\log_{4}(4^{x}+1) - \frac{1}{2}x = \frac{1}{2}x + a$ has no real roots, find the range of values for $a$;  
(3) If the function $h(x) = 4^{f(x)+ \frac{1}{2}x} + m \cdot 2^{x} - 1$, where $x \in [0, \log_{2}3]$, 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, explain why.

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