
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
Tetris is a Soviet block game developed in  $1984$ , probably to torture misbehaving middle school children. Nowadays, Tetris is a game that people play for fun, and we even have a mini-event featuring it, but it shall be used on this test for its original purpose. The  $7$  Tetris pieces, which will be used in various problems in this theme, are as follows:
![Image](https://cdn.artofproblemsolving.com/attachments/b/c/f4a5a2b90fcf87968b8f2a1a848ad32ef52010.png)**p1.** Each piece has area  $4$ . Find the sum of the perimeters of each of the  $7$  Tetris pieces.**p2.** In a game of Tetris, Qinghan places  $4$  pieces every second during the first  $2$  minutes, and  $2$  pieces every second for the remainder of the game. By the end of the game, her average speed is  $3.6$  pieces per second. Find the duration of the game in seconds.**p3.** Jeff takes all  $7$  different Tetris pieces and puts them next to each other to make a shape. Each piece has an area of  $4$ . Find the least possible perimeter of such a shape.**p4.** Qepsi is playing Tetris, but little does she know: the latest update has added realistic physics! She places two blocks, which form the shape below. Tetrominoes  $ABCD$  and  $EFGHI J$  are both formed from  $4$  squares of side length  $1$ . Given that  $CE = CF$ , the distance from point  $I$  to the line  $AD$  can be expressed as  $\frac{A\sqrt{B}-C}{D}$  . Find  $1000000A+10000B +100C +D$ .
![Image](https://cdn.artofproblemsolving.com/attachments/9/a/5e96a855b9ebbfd3ea6ebee2b19d7c0a82c7c3.png)**p5.** Using the following tetrominoes:
![Image](https://cdn.artofproblemsolving.com/attachments/3/3/464773d41265819c4f452116c1508baa660780.png)
Find the number of ways to tile the shape below, with rotation allowed, but reflection disallowed: 
![Image](https://cdn.artofproblemsolving.com/attachments/d/6/943a9161ff80ba23bb8ddb5acaf699df187e07.png)


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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