
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
Let $x$ and $y$ satisfy the constraints:
\[
\begin{cases}
x + 2y - 1 \geqslant 0, \\
x - y \geqslant 0, \\
0 \leqslant x \leqslant k,
\end{cases}
\]
If the minimum value of $z = x + ky$ is $-2$, then the maximum value of $z$ is ______.

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