
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
Find the point estimates of the unknown parameters $\alpha$ and $\beta$ of the gamma distribution using the method of moments from the sample $x_{1}, x_{2}, \ldots, x_{n}$, whose density function is given by

$$
f(x)=\frac{1}{\beta^{\alpha+1} \Gamma(\alpha+1)} x^{\alpha} e^{-x / \beta} \quad(\alpha>-1, \beta>0, x \geqslant 0)
$$

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