
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 $f(x)=\log_{2}x$:
$(1)$ Solve the inequality with respect to $x$, $f(x+1)-f(x) > 1$;
$(2)$ Let function $g(x)=f(2^{x}+1)+kx$, if the graph of $g(x)$ is symmetric about the y-axis, find the value of the real number $k$.

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