
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 $f(x)=\log_{2}(4^x+1)-kx$ $(k\in \mathbb{R})$.

$(1)$ If $f(x)$ is an even function, find the value of the real number $k$;

$(2)$ Let $g(x)=f(x)-a+1$, $k=2$, if the function $g(x)$ has a zero, find the range of values for $a$;

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