
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 $x$ and $y$ be two distinct positive integers. Define three sequences $(A_n),$ $(G_n),$ and $(H_n)$ as follows: First, $A_1 = \frac{x^2 + y^2}{2},$ $G_1 = \sqrt{xy},$ and $H_1 = \frac{2xy}{x+y}.$ Then for $n \ge 2,$ $A_n,$ $G_n,$ and $H_n$ are defined as the arithmetic mean, geometric mean, and harmonic mean of $A_{n-1}$ and $H_{n-1},$ respectively.

Consider the following statements:
1. $A_1 > A_2 > A_3 > \dotsb.$
2. $A_1 = A_2 = A_3 = \dotsb.$
4. $A_1 < A_2 < A_3 < \dotsb.$
8. $G_1 > G_2 > G_3 > \dotsb.$
16. $G_1 = G_2 = G_3 = \dotsb.$
32. $G_1 < G_2 < G_3 < \dotsb.$
64. $H_1 > H_2 > H_3 > \dotsb.$
128. $H_1 = H_2 = H_3 = \dotsb.$
256. $H_1 < H_2 < H_3 < \dotsb.$

Enter the labels of the statements that must hold.

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