
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 $C: x^2 = 2py$ ($p > 0$) with focus $F$, the line $x = 4$ intersects the x-axis at point $P$ and intersects $C$ at point $Q$, and $|QF| = \frac{5}{4}|PQ|$.  
(Ⅰ) Find the equation of $C$;  
(Ⅱ) If point $A(-a, a)$ ($a > 0$) is on the parabola $C$, is there a line $l: y = kx + 4$ that intersects $C$ at points $M$ and $N$, such that $\triangle MAN$ is a right-angled triangle with $MN$ as the hypotenuse? If it exists, find the equation of line $l$; if not, explain why.

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