
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 point (x, y) is a moving point within the plane region represented by the inequality group:

$$\begin{cases} x \geq 1 \\ x + y \leq 4 \\ ax + by + 2 \geq 0 \end{cases}$$

And the maximum value of the objective function $z = 2x + y$ is 7, while the minimum value is 1, find the value of $\frac{a}{b}$.

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