
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 $C: x^{2}=2py\left(p \gt 0\right)$, point $P(2$,$y_{0})$ lies on the parabola $C$ and is at a distance of $2$ from the focus $F$. <br/>$(1)$ Find the equation of the parabola $C$ and its directrix equation; <br/>$(2)$ Given $M\left(2,-1\right)$, the line $y=kx+1\left(k\neq 0\right)$ intersects the parabola $C$ at points $A$ and $B$, with the slopes of lines $MA$ and $MB$ denoted as $k_{1}$ and $k_{2}$ respectively. Find the value of $\frac{1}{{{k_1}}}+\frac{1}{{{k_2}}}$.

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