
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 numbers \( x_{1}, \ldots, x_{n} \) within the interval \(\left[0, \frac{\pi}{2}\right]\), find the maximum value of the expression

$$
A=\left(\sqrt{\sin x_{1}}+\ldots+\sqrt{\sin x_{n}}\right) \cdot\left(\sqrt{\cos x_{1}}+\ldots+\sqrt{\cos x_{n}}\right).
$$

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