
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 $f(x)=\sqrt{3}\sin x\cos x+\cos^2x$, the graph of function $f(x)$ is translated to the right by $\dfrac{\pi}{6}$ units and then the horizontal coordinate is expanded to twice its original size to obtain the graph of function $g(x)$. Determine the range of values for the real number $k$ such that the equation $g(x)-k=0$ has two different real roots when $x\in[0,\pi]$.

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