
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 the line $l: y = 2x + b$ and the parabola $C: x^2 = 4y$,

$(1)$ If line $l$ is tangent to the parabola $C$, find the value of the real number $b$;

$(2)$ If line $l$ passes through the focus of the parabola and intersects the parabola at points $A$ and $B$, when a moving point $P$ on the parabola moves from $A$ to $B$, find the maximum value of the area of $\triangle ABP$.

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