
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 19th Asian Games will be held in Hangzhou, Zhejiang, China in 2023. The Hangzhou Asian Games is positioned as "China's new era, Hangzhou's new Asian Games," with the goal of "Chinese characteristics, Zhejiang style, Hangzhou charm, and wonderful diversity." It upholds the concept of "green, intelligent, thrifty, and civilized" in organizing the event and adheres to the principle of "Hangzhou as the main city, shared by the whole province of Zhejiang" in hosting the games. Before the event, to welcome the Asian Games, a certain shopping mall organized a "civilized welcome to the Asian Games" knowledge competition. Each participant needs to answer three questions, A, B, and C, to earn points and subsequently receive corresponding gifts. A wrong answer earns 0 points, a correct answer to question A earns 1 point, and a correct answer to questions B and C earns 2 points each. The final score of each participant is the cumulative score of each question. It is known that the probability of a participant answering question A correctly is $\frac{2}{3}$, and the probability of answering questions B and C correctly is $\frac{1}{3}$ each, and whether each question is answered correctly is independent of the others.  
$(1)$ Find the probability that the participant answers exactly two questions correctly.  
$(2)$ Find the probability distribution and mathematical expectation of the final cumulative score of the participant.

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