
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
Let \( u \) and \( v \) be positive real numbers. For a given positive integer \( n \), find the necessary and sufficient conditions that \( u \) and \( v \) must satisfy such that there exist real numbers \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \geqslant 0 \) meeting the following conditions:
$$
\begin{array}{l}
a_{1}+a_{2}+\cdots+a_{n}=u, \\
a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=v.
\end{array}
$$

When such numbers exist, find the maximum and minimum values of \( a_{1} \).

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