
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 point $A({1,\sqrt{2}})$ lies on the curve $E:2mx^{2}+my^{2}=1$, the line $l$ with slope $\sqrt{2}$ intersects the curve $E$ at points $B$ and $C$, where $B$ and $C$ are distinct from point $A$. The following conclusions are made:<br/>$(1)$ The curve $E$ has two foci with coordinates $({-\sqrt{2},0})$ and $({\sqrt{2},0})$;<br/>$(2)$ If all the points on the curve $E$ have their x-coordinates enlarged by a factor of $\sqrt{2}$ (while keeping the y-coordinates unchanged), the resulting curve is a circle;<br/>$(3)$ The maximum area of triangle $ABC$ is $\sqrt{2}$;<br/>$(4)$ The maximum length of segment $BC$ is $3$.<br/>The correct conclusions are represented by the following numbers: ____.

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