
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 curve $E$, the distance from a point on the curve to the line $y=-2$ is $1$ greater than its distance to the point $F(0,1)$.  
$(1)$ Find the equation of curve $E$;  
$(2)$ If a chord $AB$ of curve $E$ passes through $M(1,4)$ and $M$ is the midpoint of $AB$, find the equation of the line on which chord $AB$ lies.

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