
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]


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The random variable $X$ (the water level in a river compared to the nominal level) is governed by a gamma distribution with density defined by the parameters $\alpha$ and $\beta$ $(\alpha>-1, \beta>0)$:

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

Below is the distribution of the average water level across $n=45$ floods (the first row contains the average water level $x_{i}$ (cm); the second row contains the frequency $n_{i}$ - the number of floods with an average water level of $x_{i}$):

$$
\begin{array}{cccccccccc}
x_{i} & 37.5 & 62.5 & 87.5 & 112.5 & 137.5 & 162.5 & 187.5 & 250 & 350 \\
n_{i} & 1 & 3 & 6 & 7 & 7 & 5 & 4 & 8 & 4
\end{array}
$$

Find the method of moments point estimates for the unknown parameters $\alpha$ and $\beta$ of the given gamma distribution.

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