
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
**N**ew this year at HMNT: the exciting game of RNG baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number n uniformly at random from  $\{0, 1, 2, 3, 4\}$ . Then, the following occurs:
• If  $n>0$ , then the player and everyone else currently on the field moves (counterclockwise) around
the square by n bases. However, if in doing so a player returns to or moves past the home base,
he/she leaves the field immediately and the team scores one point.
• If  $n=0$  (a strikeout), then the game ends immediately; the team does not score any more points.
What is the expected number of points that a given team will score in this game?

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