
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 $f(x)=|x^2-1|+x^2+kx$, with the domain $(0,2)$.

$(1)$ Solve the equation $f(x)=kx+3$ in the interval $(0,2)$.

$(2)$ If $f(x)$ is a monotonically decreasing function in the interval $(0,2)$, find the range of the real number $k$.

$(3)$ If the equation $f(x)=0$ has two different real roots in the interval $(0,2)$, find the range of the real number $k$.

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