
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
Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the top vertex of the uppermost cube, determine the length of the portion of $\overline{XY}$ contained in the cube with edge length $4$.

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