
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 real numbers $x$ and $y$ satisfy the conditions
$$
\begin{cases}
\begin{matrix}
x+y\leqslant 10 \\
3x+y\leqslant 18
\end{matrix} \\
\begin{matrix}
x\geqslant 0 \\
y\geqslant 0
\end{matrix}
\end{cases}
$$
then the maximum value of $z=x+ \frac{y}{2}$ is __________.

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