
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 $f(x)=\log _{4}\left(4^{x}+1\right)+kx$, where $k\in R$, and its graph is symmetric about the y-axis.

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

(2) If the equation concerning $x$, $\log _{4}\left(4^{x}+1\right)- \frac {1}{2}x= \frac {1}{2}x+a$ has no real roots, find the range of values for $a$;

(3) If the function $h(x)=4\;^{f(x)+ \frac {1}{2}x}+m\cdot2^{x}-1$, $x\in[0,\log _{2}3]$, 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 it does not exist, explain the reason.

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