
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
Given the joint density function of the continuous two-dimensional random variable (X, Y):

\[ f(x, y)= 
\begin{cases}
4 x y e^{-x^{2}-y^{2}} & (x>0, y>0) \\
0 & (x<0 \, \text{or} \, y<0).
\end{cases}
\]

Find: 
a) Expectations (means) of X and Y. 
b) Variances of X and Y.

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