
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
A and B each roll a six-sided die (its six faces are marked with the numbers $1$, $2$, $3$, $4$, $5$, $6$) once. Let the numbers on the top faces of the dice thrown by A and B be $x$ and $y$ respectively. Then:

$(1)$ How many different outcomes are there?

$(2)$ List all the outcomes where the real part of the complex number $x+yi$ is greater than its imaginary part;

$(3)$ What is the probability that the real part of the complex number $x+yi$ is greater than its imaginary part?

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