
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 $f(x)$ satisfies $f(x+y)-f(y)=(x+2y-2)x$ for all real numbers $x$, $y$, and $f(1)=0$.

$(1)$ Find the expression for the function $f(x)$;  
$(2)$ Let $g(x)=\frac{f(x)-2x}{x}$, if the inequality $g(2^x)-k\cdot 2^x\leqslant 0$ ($k$ is a constant) always holds for $x\in [-2,2]$, find the range of the real number $k$.

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