
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 real numbers, and \( k \) a positive real number different from 1. Define three sequences \( (A_n) \), \( (G_n) \), and \( (H_n) \) as follows: Firstly, \( A_1 = \frac{kx + y}{k+1} \), \( G_1 = \sqrt[k+1]{x^k \cdot y} \), and \( H_1 = \frac{(k+1)xy}{kx+y} \). For \( n \geq 2 \), \( A_n \), \( G_n \), and \( H_n \) are the arithmetic mean, geometric mean, and harmonic mean of \( A_{n-1} \) and \( H_{n-1} \), respectively.

Analyze the following statements to determine which are true:
1. \( A_1 > A_2 > A_3 > \dots \)
2. \( A_1 = A_2 = A_3 = \dots \)
4. \( A_1 < A_2 < A_3 < \dots \)
8. \( G_1 > G_2 > G_3 > \dots \)
16. \( G_1 = G_2 = G_3 = \dots \)
32. \( G_1 < G_2 < G_3 < \dots \)
64. \( H_1 > H_2 > H_3 > \dots \)
128. \( H_1 = H_2 = H_3 = \dots \)
256. \( H_1 < H_2 < H_3 < \dots \)

Sum the labels of the true statements.

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