
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 that line $l$, which passes through point $A(-4,0)$ with a slope of $\frac{1}{2}$, intersects the parabola $G:x^2=2py (p>0)$ at points $B$ and $C$. The x-coordinate of the midpoint of segment $BC$ is $1$.

(1) Find the equation of parabola $G$;

(2) Let the y-intercept of the perpendicular bisector of segment $BC$ be $b$, find the value of $b$.

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