
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
The 22nd Asian Games will be held in Hangzhou, China from September 23 to October 8, 2023. This is the second time China has hosted the Asian Games after Beijing. To welcome this sports event, a city in Zhejiang Province decided to hold an Asian Games knowledge competition. Community A in the city held a selection competition, which was divided into preliminary and final rounds. Only those who passed the preliminary round could participate in the final round. Those who passed the final round would represent Community A in the city's Asian Games knowledge competition. It is known that three players, A, B, and C from Community A participated in the preliminary round, and the probabilities of them passing the preliminary round are $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{2}$ respectively. The probability of passing the final round after passing the preliminary round is $\frac{1}{3}$ for each of them. Assume that their passing or not passing does not affect each other.  
$(1)$ Find the probability that at least one of these three players will participate in the city's knowledge competition.  
$(2)$ A brand sponsor supported this knowledge competition in Community A and provided two reward options for the players participating in the selection competition:  
Option 1: All players participating in the selection competition can participate in a lottery. Each person gets one chance, and the probability of winning each time is $\frac{1}{3}$. The winnings do not affect each other. Winning once rewards $600.  
Option 2: Players who only participated in the preliminary round are rewarded $100, and players who participated in the final round are rewarded $400 (including the $100 for participating in the preliminary round). If the brand sponsor wants to give more rewards to the players, analyze from the perspective of the mathematical expectation of the total prize money for the three players, which option is better.

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