
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 hyperbola $C:\frac{{{y^2}}}{{{a^2}}}-\frac{{{x^2}}}{{{b^2}}}=1(a＞0,b＞0)$ with an asymptote equation $y=2x$, and a distance of $1$ from one focus to the asymptote line.<br/>$(1)$ Find the equation of $C$;<br/>$(2)$ A line $l$ passing through point $M\left(1,4\right)$ intersects $C$ at points $A$ and $B$, where $M$ is the midpoint of segment $AB$. Find the equation of $l$.

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