
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 game activity of a television entertainment program, each person needs to complete three tasks: $A$, $B$, and $C$. It is known that the probabilities of contestant Jia completing tasks $A$, $B$, and $C$ are $\frac{3}{4}$, $\frac{3}{4}$, and $\frac{2}{3}$, respectively. Each task is independent of the others.

(1) If contestant Jia attempts tasks $A$, $B$, and $C$ once each, find the probability that Jia completes at least one task.

(2) The activity requires tasks $A$ and $B$ to be attempted twice each and task $C$ to be attempted three times. If both attempts of task $A$ are successful, then task $B$ is attempted, and points $a$ are awarded. If both attempts of task $B$ are successful, then task $C$ is attempted, and points $3a$ are awarded. If out of three attempts at task $C$, at least two are successful, then the contestant succeeds and is awarded 6a points (points are not cumulative). Each task is independent of the others. Let $X$ represent the number of points contestant Jia earns. Write the distribution of $X$ and find the expected value.

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