
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
In a certain television entertainment game show, each participant needs to complete three events $A$, $B$, and $C$. It is known that the probabilities for contestant Jia to complete events $A$, $B$, and $C$ are $\frac{3}{4}$, $\frac{3}{4}$, and $\frac{2}{3}$ respectively. Each event is mutually independent. 

(1) Contestant Jia attempts each of the events $A$, $B$, and $C$ once. Find the probability that Jia completes at least one of the events.

(2) The activity requires events $A$ and $B$ to be attempted twice each, and event $C$ to be attempted three times. If both attempts of event $A$ are completed, then proceed to event $B$, and earn $a$ points; if both attempts of event $B$ are completed, then proceed to event $C$, and earn $3a$ points; and if event $C$ is completed at least two out of three times, then the contestant successfully completes the challenge and earns $6a$ points (points are not cumulative). Each event is mutually independent. Let $X$ represent the number of points contestant Jia earns. Write the probability distribution of $X$ and calculate the expected value of $X$.

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