
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 $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, the intersection point of the line $x=4$ with the $x$-axis is $P$, and with $C$ is $Q$, and $|QF|=\frac{5}{4}|PQ|$. <br/>$(Ⅰ)$ Find the equation of $C$; <br/>$(Ⅱ)$ Point $A\left(-a,a\right)\left(a \gt 0\right)$ lies on the parabola $C$. Does there exist 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 the line $l$; if not, explain the reason.

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