
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 $C_1$: $\dfrac{y^2}{a^2} + \dfrac{x^2}{b^2} = 1$ ($a > b > 0$) with one focus at $F_1(0, \sqrt{5})$, and passing through a point $P\left(\dfrac{4}{3}, \sqrt{5}\right)$.
$(I)$ Find the standard equation of the ellipse $C_1$;
$(II)$ Given another ellipse $C_2$ whose center is at the origin, with foci on the $y$-axis, and whose major and minor axes are $\lambda$ times the length of the major and minor axes of $C_1$ ($\lambda > 1$), a line $l$ passing through point $C(-1,0)$ intersects $C_2$ at two distinct points $A$ and $B$. If $\overrightarrow{AC} = 2 \overrightarrow{CB}$, find the equation of the line $l$ when the area of $\triangle OAB$ is maximized.

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