
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 Li's father bought a ticket for a concert, and both Xiao Li and her brother wanted to go watch it. However, there was only one ticket available. Xiao Li's ninth-grade brother came up with a solution. He took eight playing cards and gave Xiao Li the cards with the numbers $1$, $2$, $3$, and $5, while keeping the cards with the numbers $4$, $6$, $7$, and $8 for himself. They played a game according to the following rules: Xiao Li and her brother each randomly drew one card from their respective sets of four cards, added the numbers on the two cards, and if the sum was even, Xiao Li would go; if the sum was odd, her brother would go.<br/>$(1)$ Please use a tree diagram or list to represent all possible results of the sum of the two card numbers.<br/>$(2)$ Is the game rule designed by Xiao Li's brother fair? Why? If not fair, please design a fair game rule.

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