
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 a parabola $y=ax^2+bx$ that is tangent to the line $x+y=4$ within the first quadrant. Let $S$ be the area enclosed by this parabola and the x-axis. Find the values of $a$ and $b$ that maximize $S$, and determine the maximum value of $S$.

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