
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 variables $x$ and $y$ that satisfy the system of inequalities $\begin{cases} & x-2y+2 \geqslant 0, \\ & 2x-y-2 \leqslant 0, \\ & y \geqslant 0, \end{cases}$ find the minimum value of $z=3x+y$.

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