
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
There are two parabolas $y = x^2 - 3x$ and $y = -x^2 + 9$. A line passing through point $P(t, 0)$ and parallel to the y-axis intersects these two parabolas at points $A$ and $B$. When $t$ varies within the range from $0$ to $3$, find the maximum value of the length of segment $AB$.

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