
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 $y$ be real numbers that satisfy the system of inequalities $\begin{cases} \begin{matrix} x - y + 1 \geqslant 0 \\ x + y - 1 \geqslant 0 \\ 3x - y - 3 \leqslant 0 \end{matrix} \end{cases}$. Find the sum of the maximum and minimum possible values of $z = |x - 4y + 1|$.

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