
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 parabola $C_1: x^2 = 2py$ whose focus is on the parabola $C_2: y = x^2 + 1$, and point $P$ is a moving point on parabola $C_1$.

$(1)$ Find the equation of parabola $C_1$ and its directrix equation;

$(2)$ Draw two tangents from point $P$ to parabola $C_2$, with $A$ and $B$ being the points of tangency, find the minimum value of the area of $\Delta PAB$.

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