
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
Calculate the double integral

$$
\iint_{D} x \, dx \, dy
$$

where the region \( D \) is bounded by the lines

$$
y^{2} - 4y + x^{2} = 0, \quad y^{2} - 8y + x^{2} = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0
$$

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