
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 an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ ($a > b > 0$) with a focal length of $4$, and passing through point $P(2, \frac{5}{3})$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ If a line $l$ passes through $M(0, 1)$ and intersects $C$ at points $A$ and $B$, and $\vec{MA} = - \frac{2}{3}\vec{MB}$, find the equation of the line $l$.

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