
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
Four chess players - Ivanov, Petrov, Vasiliev, and Kuznetsov - played a round-robin tournament (each played one game against each of the others). A victory awards 1 point, a draw awards 0.5 points to each player. It was found that the player in first place scored 3 points, and the player in last place scored 0.5 points. How many possible distributions of points are there among the named chess players, if some of them could have scored the same number of points? (For example, the cases where Ivanov has 3 points and Petrov has 0.5 points, and where Petrov has 3 points and Ivanov has 0.5 points, are considered different!)

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