
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
There is one movie ticket, and Mingming and Leilei plan to decide who owns it by playing a dice game. The rules of the game are as follows: Each face of a fair cubic die is labeled with the numbers $1$, $2$, $3$, $4$, $5$, $6$. Mingming and Leilei each roll a die once. If the sum of the two numbers facing up is a multiple of $3$, then Mingming wins and gets the movie ticket; otherwise, Leilei wins and gets the movie ticket.<br/>$(1)$ The probability that Mingming rolls the die once and the number facing up is a multiple of $3$ is______.<br/>$(2)$ According to the rules of the game, who do you think has a higher chance of winning, Mingming or Leilei? Please explain your reasoning using a list or a tree diagram.

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