
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
Square $ABCD$ has a side length of $3$. Point $E$ is on side $AB$ such that $AE = 1$. Two players, Alpha and Beta, play the following game: Alpha first chooses a point $X$ on line $BE$, then Beta chooses a point $Y$ on side $BC$. The square is then folded along $DX$ and $DY$ such that points $A$ and $C$ coincide at point $A'$. Alpha's objective is to make the angle between $A'D$ and the plane $DXY$ as small as possible, while Beta's objective is to make the same angle as large as possible. Assuming both players use optimal strategies, find the size of the angle between $A'D$ and plane $DXY$, and determine the positions of points $X$ and $Y$.

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