
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 real numbers $a$ and $b$ satisfy $2a+b=9$.  
$(i)$ If $|9-b|+|a| < 3$, find the range of values for $x$.  
$(ii)$ If $a$, $b > 0$, and $z=a^{2}b$, find the maximum value of $z$.

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