
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 that real numbers \( x \) and \( y \) satisfy the constraints
\[
\begin{cases}
x + 2y > 2 \\
2x + y \leq 4 \\
4x - y \geq 1
\end{cases}
\]
what is the range of values that the objective function \( z = 3x + y \) can take?

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