
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
<u>Set 3</u>**p7.** On unit square  $ABCD$ , a point  $P$  is selected on segment  $CD$  such that  $DP =\frac14$  . The segment  $BP$  is drawn and its intersection with diagonal  $AC$  is marked as  $E$ . What is the area of triangle  $AEP$ ?**p8.** Five distinct points are arranged on a plane, creating ten pairs of distinct points. Seven pairs of points are distance  $1$  apart, two pairs of points are distance  $\sqrt3$  apart, and one pair of points is distance  $2$  apart. Draw a line segment from one of these points to the midpoint of a pair of these points. What is the longest this line segment can be?**p9.** The inhabitants of Mars use a base  $8$  system. Mandrew Mellon is competing in the annual Martian College Interesting Competition of Math (MCICM). The first question asks to compute the product of the base  $8$  numerals  $1245415_8$ ,  $7563265_8$ , and  $ 6321473_8$ . Mandrew correctly computed the product in his scratch work, but when he looked back he realized he smudged the middle digit. He knows that the product is  $1014133027\blacksquare 27662041138$ . What is the missing digit?


PS. You should use hide for answers.

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