
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
Laser beams are known for reflecting off solid objects. Whenever a laser beam hits a straight, solid wall, it reflects off in the opposite direction, at an angle to the wall that is equal to the angle at which it hits, as shown.
![Image](https://cdn.artofproblemsolving.com/attachments/8/2/e56b311d1cea61b999d36cbe9189b84586a70b.png)

The path of the laser  $L$  meets the wall,  $W$ , at an angle of  $55.15$  degrees coming from the right. Then,  $L$  reflects off  $W$  to a redirected path,  $L'$  , at an angle of  $55.15$  degrees going off to the left. In particular, if  $L$  were to meet  $W$  at an angle of  $90$  degrees, then  $L' $  would follow  $L$ , going backwards (straight up).**p6.** Given a square  $ABCD$ , with  $AB = 1$ , mark the midpoints  $M$  and  $N$  of  $AB$  and  $BC$ , respectively. A laser beam shot from  $M$  to  $N$ , and the beam reflects of  $BC$ ,  $CD$ ,  $DA$ , and comes back to  $M$ . This path encloses a smaller area inside square  $ABCD$ . Find this area.**p7.** Given a rectangle  $EFGH$ , with  $EF = 3$  and  $FG = 40$ , mark a point  $P$  on  $FG$  such that  $FP = 4$ . A laser beam is shot from  $E$  to  $P$ , which then reflects off  $FG$ , then  $EH$ , then  $FG$ , etc. Once it reaches some point on  $GH$ , the beam is absorbed; it stops reflecting. How far does the beam travel?**p8**. Given a rectangle  $EF GH$  with  $EF = 3$  and  $FG = 2010$ , mark a point  $P$  on  $F G$  such that  $FP = 4$ . A laser beam is shot from  $E$  to  $P$ , which then reflects off  $FG$ , then  $EH$ , then  $FG$ , etc. Once it reaches some point on  $GH$ , the beam is absorbed; it stops reflecting. How far does the beam travel?**p9.** Given a triangle  $XYZ$  with  $\angle Y = 90^o$ ,  $XY = 1$ , and  $XZ = 2$ , mark a point  $Q$  on  $YZ$  such that   $\frac{ZQ}{ZY} = \frac13$ . A laser beam is shot from  $Q$  perpendicular to  $YZ$ , and it reflects off the sides of  $XYZ$  indefinitely. How many bounces does it take for the laser beam to get back to  $Q$  for the first time (not including the release from  $Q$  and the return to  $Q$ )?**p10.** Given a triangle  $XYZ$  with  $\angle Y = 90^o$ ,  $XY = 1$ , and  $XZ = 2$ , mark a point  $Q$  on  $YZ$  such that  $\frac{ZQ}{ZY} = \frac13$ . A laser beam is shot from  $Q$  perpendicular to  $YZ$ , and it reflects off the sides of  $XYZ$  indefinitely. How far has the laser traveled when it reaches its  $2010$ th bounce?


PS. You should use hide for answers.

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