
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 $a$ and $b$ are two distinct positive numbers, insert two sets of numbers $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_n$ ($n \in \mathbb{N}^+$, and $n \geq 2$) between $a$ and $b$, making $a, x_1, x_2, \ldots, x_n, b$ an arithmetic sequence and $a, y_1, y_2, \ldots, y_n, b$ a geometric sequence. Among the following four expressions, the one(s) that must hold is/are \_\_\_\_\_\_. (Fill in the blank with the correct option number(s).)  
① $$\sum\limits_{k=1}^{n}x_{k}= \frac {n(a+b)}{2}$$; ② $$\frac {1}{n} \sum\limits_{k=1}^{n}x_{k} = \frac {a+b}{2} > \sqrt {ab}+ \left( \frac { \sqrt {a}- \sqrt {b}}{2}\right)^{2}$$; ③ $$ny_{1}y_{2}\ldots y_{n} = \sqrt {ab}$$; ④ $$ny_{1}y_{2}\ldots y_{n} > \frac {2ab}{a+b}$$.

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