
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 that the center of the ellipse $E$ is at the origin, the focus is on the $y$-axis with an eccentricity of $e=\frac{\sqrt{2}}{2}$, and it passes through the point $P\left( 1,\sqrt{2} \right)$:

(1) Find the equation of the ellipse $E$;

(2) Two pairs of perpendicular lines are drawn through the focus $F$ of ellipse $E$, intersecting the ellipse $E$ at $A,B$ and $C,D$ respectively. Find the minimum value of $| AB |+| CD |$.

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