
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 school plans to hold a fun sports competition, which consists of several traditional and new events. Each athlete needs to participate in $3$ sports events in total. For each traditional event, if not completed, the athlete gets $0$ points; if completed, the athlete gets $30$ points. For new events, if not completed, the athlete gets $0$ points; if only one is completed, the athlete gets $40$ points, and if two are completed, the athlete gets $90$ points. The more points the athlete scores, the more bonus they will receive. There are two participation options available for athletes:
- Option 1: Participate in $3$ traditional events;
- Option 2: Participate in $1$ traditional event and $2$ new events;
Assume that the probability of completing each traditional event is $\frac{1}{2}$, and the probability of completing each new event is $\frac{1}{3}$, and the completion of each event by the athlete is independent.
$(1)$ If an athlete chooses Option 1, let the final score be $X$, find the distribution and expectation of $X$;
$(2)$ Based on the expected value of the final score, which participation option should the athlete choose? Explain your reasoning.

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