
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 $C: y^2 = 2px (p > 0)$ with focus $F$ that coincides with one focus of the ellipse $C': \frac{x^2}{6} + \frac{y^2}{5} = 1$. Point $A(x_0, 2)$ lies on the parabola, and the line $l$ passing through focus $F$ intersects the parabola at points $M$ and $N$.

(1) Find the equation of parabola $C$ and the value of $|AF|$;
(2) Let the directrix of parabola $C$ intersect the $x$-axis at point $B$. If $\overrightarrow{MF} = \lambda \overrightarrow{FN}$ and $|BM|^2 + |BN|^2 = 40$, find the value of the real number $\lambda$.

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