
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 line $l$ and an ellipse $E$: $\dfrac{y^2}{8} + \dfrac{x^2}{4} = 1$ have two distinct intersection points $A$ and $B$, and the midpoint of chord $AB$ is point $P(1,2)$;

$(1)$ Find the equation of line $l$;

$(2)$ If the function $f(x) = x^2 - ax + b$ has its tangent line at point $P(1,2)$ exactly as line $l$; find the expression of $f(x)$.

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