
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 the focus of the parabola $C_{1}$: $y^{2}=2px\left(p \gt 0\right)$ coincides with the right vertex of the hyperbola $C_{2}$: $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{12}=1$. <br/>$(Ⅰ)$ Find the standard equation of the parabola $C_{1}$; <br/>$(Ⅱ)$ Let the line $l$ passing through the point $\left(0,1\right)$ intersect the parabola $C_{1}$ at two distinct points $A$ and $B$. $F$ is the focus of the parabola $C_{1}$, and $\overrightarrow{FA}\cdot \overrightarrow{FB}=1$. Find the equation of the line $l$.

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