
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 $C: y^2 = 2px$ ($p > 0$) and a line $y = x + 1$ that are tangent to each other.

1. Find the equation of the parabola $C$.
2. Let $A(x_1, y_1)$ and $B(x_2, y_2)$ be two moving points on curve $C$, where $x_1 \neq x_2$ and $x_1 + x_2 = 4$. The line $l$, which is the perpendicular bisector of segment $AB$, intersects the x-axis at point $Q$. Find the maximum value of the area of $\triangle ABQ$.

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