
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
Xiao Wen and Xiao Li drew a straight line on the paper, and played a game of "rock-paper-scissors, roll the dice, move the chess piece" on this line. Each of them holds a chess piece (they move their own chess pieces and do not affect each other). The winner of the rock-paper-scissors game rolls the dice first, and the number on the dice determines how many centimeters the chess piece moves eastward; the loser of the rock-paper-scissors game rolls the dice afterwards, and the number on the dice determines how many centimeters the chess piece moves westward. Now, both of them start at point $A$ on the line and play the game. It is defined that moving eastward is positive and moving westward is negative. Below is the record of five consecutive games played by Xiao Wen and Xiao Li (starting from the second game, the starting point of the chess piece is the endpoint of the previous game): (unit: centimeters) (In case of a tie in rock-paper-scissors, they re-play until a winner is determined)

|         | 1st Game | 2nd Game | 3rd Game | 4th Game | 5th Game |
|---------|----------|----------|----------|----------|----------|
| Xiao Wen| $+3$     | $-2$     | $-1$     | $+4$     | $-5$     |
| Xiao Li | $-4$     | $+3$     | $+5$     | $-2$     | $a$      |

$(1)$ At the end of the fifth game, in which direction is Xiao Wen's chess piece from point $A$? How far is it from point $A$?  
$(2)$ At the end of the fifth game, Xiao Li's chess piece is 5 centimeters east of point $A$. Find the value of $a$.  
$(3)$ Under the condition in $(2)$, Xiao Wen scores 3 points for every centimeter moved eastward and loses 2 points for every centimeter moved westward. At the end of the fifth game, how many more points does the player with the higher score have compared to the player with the lower score?

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