
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 $f(x) = 0.5x$ in the interval $(0, 2)$, and $f(x) = 0$ outside this interval. Find the raw moments (moments about the origin) and central moments of the first, second, third, and fourth orders.

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