
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
Suppose $x$ and $y$ satisfy the linear constraints $\begin{cases} 2x - y + 2 \geqslant 0 \\ x - 3y + 1 \leqslant 0 \\ x + y - 2 \leqslant 0 \end{cases}$, and the maximum value of $z = ax - y (a > 0)$ has multiple optimal solutions. Find the value of the real number $a$.

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