
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
On a rectangular billiard table $ABCD$, a ball is hit from corner $A$ in the direction of the angle bisector, striking sides $CD$, $BC$, and then $AB$ before hitting a ball positioned exactly at the center of the rectangle. In what other direction can we hit a ball from point $A$ such that, after bouncing off three different sides, it also hits the ball placed at the center of the rectangle?

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