
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 $\mathbf{C}$ be a subset of $\mathbf{R}^{+}$, and let $u$ and $v$ be positive real numbers. Determine the necessary and sufficient conditions that $u$ and $v$ must satisfy, so that, for a given $n$, there exist real numbers $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \geqslant 0$ satisfying 
$$
a_{1}+a_{2}+\cdots+a_{n}=u \quad \text{and} \quad a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=v.
$$

Find the maximum and minimum values of $a_{1}$.

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