
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
p1. Known points  $A (-1.2)$ ,  $B (0,2)$ ,  $C (3,0)$ , and  $D (3, -1)$  as seen in the following picture.
 Determine the measure of the angle  $AOD$  .
![Image](https://cdn.artofproblemsolving.com/attachments/f/2/ca857aaf54c803db34d8d52505ef9a80e7130f.png)


p2. Determine all prime numbers  $p> 2$  until  $p$  divides  $71^2 - 37^2 - 51$ .


p3. A ball if dropped perpendicular to the ground from a height then it will bounce back perpendicular along the high third again, down back upright and bouncing back a third of its height, and next. If the distance traveled by the ball when it touches the ground the fourth time is equal to  $106$  meters. From what height is the ball was dropped?


p4. The beam  $ABCD.EFGH$  is obtained by pasting two unit cubes  $ABCD.PQRS$  and  $PQRS.EFGH$ . The point K is the midpoint of the edge  $AB$ , while the point  $L$  is the midpoint of the edge  $SH$ . What is the length of the line segment  $KL$ ?


p5. How many integer numbers are no greater than  $2004$ , with  remainder  $1$  when divided by  $2$ , with  remainder  $2$  when divided by  $3$ , with  remainder  $3$  when divided by  $4$ , and with  remainder  $4$  when divided by  $5$ ?

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