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:
[img]https://cdn.artofproblemsolving.com/attachments/b/c/f4a5a2b90fcf87968b8f2a1a848ad32ef52010.png[/img]

[b]p1.[/b] Each piece has area $4$. Find the sum of the perimeters of each of the $7$ Tetris pieces.


[b]p2.[/b] 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.


[b]p3.[/b] 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.


[b]p4.[/b] 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 $1000000A+10000B +100C +D$.
[img]https://cdn.artofproblemsolving.com/attachments/9/a/5e96a855b9ebbfd3ea6ebee2b19d7c0a82c7c3.png[/img]

[b]p5.[/b] Using the following tetrominoes:
[img]https://cdn.artofproblemsolving.com/attachments/3/3/464773d41265819c4f452116c1508baa660780.png[/img]
Find the number of ways to tile the shape below, with rotation allowed, but reflection disallowed: 
[img]https://cdn.artofproblemsolving.com/attachments/d/6/943a9161ff80ba23bb8ddb5acaf699df187e07.png[/img]


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to find the number of ways to tile the given shape using the provided tetrominoes, with rotation allowed but reflection disallowed. We will use a recursive approach to solve this problem.

1. **Define the Problem Recursively:**
   Let \( a_n \) be the number of ways to tile the shape formed by attaching a \( 1 \times 1 \) square to the left of the bottom-left-most square and to the right of the top-right-most square of a \( 2 \times n \) grid (where \( n \) is a positive integer) using the given tetrominoes.

   Similarly, let \( b_n \) be the number of ways to tile the shape formed by attaching a \( 1 \times 1 \) square to the left of the bottom-left-most square and to the right of the top-right-most square of a \( 2 \times n \) grid (where \( n \) is a positive integer) using the given tetrominoes.

2. **Establish the Recurrence Relations:**
   We wish to find \( b_{11} \). Taking cases on what tile occupies the left-most squares of both figures, we have:
   \[
   a_n = a_{n-2} + b_{n-2}
   \]
   \[
   b_n = a_{n-2} + b_{n-2}
   \]
   So it follows that:
   \[
   a_n = b_n \quad \forall \quad n \in \mathbb{Z}_{\geq 3}
   \]

3. **Initial Conditions:**
   It is easy to find that:
   \[
   a_1 = b_1 = 1
   \]
   \[
   a_2 = b_2 = 0
   \]

4. **Simplify the Recurrence:**
   Since \( a_n = b_n \) for all \( n \geq 3 \), we can simplify the recurrence relation to:
   \[
   b_n = 2b_{n-2} \quad \forall \quad n \in \mathbb{Z}_{\geq 2}
   \]

5. **Calculate \( b_{11} \):**
   Using the recurrence relation, we can calculate \( b_{11} \) as follows:
   \[
   b_{11} = 2b_9 = 4b_7 = 8b_5 = 16b_3 = 32b_1 = 32 \cdot 1 = 32
   \]

The final answer is \( \boxed{32} \).