
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 real numbers $x$ and $y$ satisfy the conditions
\[
\begin{cases}
& y \geqslant x + 2, \\
& x + y \leqslant 6, \\
& x \geqslant 1,
\end{cases}
\]
calculate the minimum value of $z = 2|x - 2| + |y|$.

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