
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
$P$ is a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$. The intersection points of the curve $\frac{|x|}{2}+|y|=1$ with the coordinate axes are $A$, $B$, $C$, and $D$. If $|PA|+|PB|+|PC|+|PD|=4\sqrt{6}$, then the distance from $P$ to the $x$-axis is ____.

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