
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
If the objective function $z=kx+2y$ takes its minimum value only at the point $(1,1)$ under the constraints $\begin{cases} & 2x-y\leqslant 1 \\\\ & x+y\geqslant 2 \\\\ & y-x\leqslant 2 \\\\ \end{cases}$, then the range of the real number $k$ is _____.

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