
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\dfrac{1}{2}$, and a line $l$: $x + 2y = 4$ that intersects the ellipse at exactly one point $T$.

$(I)$ Find the equation of ellipse $C$ and the coordinates of point $T$;  
$(II)$ Let $O$ be the origin, and a line $l'$ parallel to $OT$ intersects the ellipse $C$ at two distinct points $A$ and $B$. Find the equation of line $l'$ when the area of $\triangle OAB$ is maximized.

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