
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 $N$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ passing through the point $C\left(0,1\right)$, and with eccentricity $\frac{\sqrt{2}}{2}$.<br/>$(1)$ Find the equation of the ellipse $N$;<br/>$(2)$ The line $l:y=kx-\frac{1}{3}$ intersects the ellipse $N$ at points $A$ and $B$, with the midpoint of the line segment $AB$ being $M$. Is there a constant $\lambda $ such that $\angle AMC=\lambda \cdot \angle ABC$ always holds true, and explain the reason.

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