
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.** An  $8$ -inch by  $11$ -inch sheet of paper is laid flat so that the top and bottom edges are  $8$  inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold?**p2.** Triangle  $ABC$  is equilateral, with  $AB = 6$ . There are points  $D$ ,  $E$  on segment AB (in the order  $A$ ,  $D$ ,  $E$ ,  $B$ ), points  $F$ ,  $G$  on segment  $BC$  (in the order  $B$ ,  $F$ ,  $G$ ,  $C$ ), and points  $H$ ,  $I$  on segment  $CA$  (in the order  $C$ ,  $H$ ,  $I$ ,  $A$ ) such that  $DE = F G = HI = 2$ . Considering all such configurations of  $D$ ,  $E$ ,  $F$ ,  $G$ ,  $H$ ,  $I$ , let  $A_1$  be the maximum possible area of (possibly degenerate) hexagon  $DEF GHI$  and let  $A_2$  be the minimum possible area. Find  $A_1 - A_2$ .**p3.** Find   $$ \tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7} $$  PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).



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