
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 random variable \(X\) is defined by the probability density function \(f(x) = a x^2\) for \(x \in (0; 1)\) and \(f(x) = 0\) for \(x \notin (0; 1)\). Determine the parameter \(a\) and the cumulative distribution function \(F(x)\). Also, find the probability that in four trials, \(X\) takes values in the interval \((0.25; 0.75)\) exactly three times.

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