
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 point P(2, 1) is on the parabola $C_1: x^2 = 2py$ ($p > 0$), and the line $l$ passes through point Q(0, 2) and intersects the parabola $C_1$ at points A and B.
(1) Find the equation of the parabola $C_1$ and the equation for the trajectory $C_2$ of the midpoint M of chord AB;
(2) If lines $l_1$ and $l_2$ are tangent to $C_1$ and $C_2$ respectively, and $l_1$ is parallel to $l_2$, find the shortest distance from $l_1$ to $l_2$.

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