
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 $a$ and $b$ be real numbers that satisfy $a + 2b = 9$.
(1) Determine the range of values for $a$ if $|9 - 2b| + |a + 1| < 3$;
(2) If $a$ and $b$ are positive and $z = ab^2$, find the maximum value of $z$.

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