
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 device generates a random 5-bit binary number $A=a\_1a\_2a\_3a\_4a\_5_{(2)}$ each time it is activated. The probability of $0$ appearing in the digits of $A$ ($a\_1=1,a\_k (k=2,3,4,5)$) is $\frac{1}{3}$, while the probability of $1$ appearing is $\frac{2}{3}$. If the generated number is $A=10101_{(2)}$, the trial is considered successful, yielding 2 points; if not, it yields -1 point. Calculate the variance of the total score $X$ after 100 repeated trials.

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