
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
Find, with proof, the smallest real number  $C$  with the following property:
For every infinite sequence  $\{x_i\}$  of positive real numbers such that  $x_1 + x_2 +\cdots + x_n \leq x_{n+1}$  for  $n = 1, 2, 3, \cdots$ , we have
\[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n}  \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.\]

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