
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
The system of inequalities 
\[
\begin{cases}
\begin{matrix}
y-1 \geqslant 0 \\
x-y+2 \geqslant 0
\end{matrix} \\
x+4y-8 \leqslant 0
\end{cases}
\]
represents a planar region denoted by $\Omega$. If the line $x=a$ ($a > 1$) divides the planar region $\Omega$ into two parts with an area ratio of $1:4$, then the maximum value of the objective function $z=ax+y$ is $\boxed{9}$.

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