
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 $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left( a\gt b\gt 0 \right)$ with eccentricity $\frac{\sqrt{5}}{3}$, a line $l$ with slope $k$ passes through the point $P(0,1)$, and intersects the ellipse $E$ at two distinct points $A$ and $B$. When the line $l$ is perpendicular to the $y$-axis, $\left| AB \right|=3\sqrt{3}$.

1. Find the standard equation of the ellipse $E$.
2. Let the coordinates of point $M$ be $\left( \frac{5}{12},0 \right)$. If $\triangle AMB$ is an isosceles triangle with $AB$ as the base, find the value of $k$.

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