
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
A rectangle of size $2021 \times 4300$ has a billiard ball located at some point inside it. The ball is launched in a straight line forming a $45^{\circ}$ angle with the sides of the rectangle. When the ball hits a side, it reflects also at a $45^{\circ}$ angle; if the ball hits a corner, it exits along the same line it entered. 

a) Is it true for any point that if a ball is released from it following these rules, it will return to that point again?

b) Suppose that starting from a certain point $A$, the ball returns to it after some time. What is the maximum number of bounces off the sides it can make before first returning to point $A$?

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