
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 $k$;  
$(2)$ Let $g(x)=\log_{4}(a\cdot 2^{x}- \frac{4}{3}a)$, if the graph of function $f(x)$ and $g(x)$ have exactly one common point, find the range of the real number $a$.

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