
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
We define a new type of triangle: a triangle in which the sum of the squares of two sides is equal to twice the square of the third side is called a "cute triangle." <br/>$(1)$① According to the definition of a "cute triangle," determine whether an equilateral triangle is ______ (fill in "is" or "is not") a cute triangle;<br/>② If the lengths of the three sides of a triangle are $4$, $2\sqrt{6}$, and $2\sqrt{5}$, then this triangle is ______ (fill in "is" or "is not") a cute triangle;<br/>$(2)$ If $Rt\triangle ABC$ is a cute triangle, $\angle C=90^{\circ}$, and $AC=2\sqrt{2}$, find the length of $AB$.

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