
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
For a random variable \( X \) defined by the distribution function \( F(x) \) as follows:
- \( F(x) = 0 \) for \( x < 2 \),
- \( F(x) = a(x-2)^2 \) for \( 2 \leq x \leq 4 \),
- \( F(x) = 1 \) for \( x > 4 \),

Determine:
1. The value of the parameter \( a \).
2. The probability density function \( f(x) \).
3. The first four raw moments and central moments.
4. The skewness and kurtosis.
5. Plot the graphs of \( f(x) \) and \( F(x) \).

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