
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 regular icosahedron is a geometric solid with 20 faces, which are equilateral triangles. By numbering the faces from 1 to 20 on a regular icosahedron, it is transformed into a die. Luísa and Mathias invented a game where each rolls the icosahedral die 5 times and records the numbers drawn in sequence, forming a single number. For example, if Luísa drew the sequence \(5, 11, 7, 20, 17\), she would note the number 51,172,017.

a) In the first round, Luísa won with the largest possible difference. What is this difference?

b) In the second round, Luísa scored 162,012,510 points and Mathias rolled the die four times, obtaining 14, 11, 8, and 19. Is it possible for Mathias to win this round with his last roll?

c) In the third round, Luísa scored 12,111,813 points. How many sequences can result in this score?

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