
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 \(a\) and \(b\) be two distinct positive real numbers not equal to each other. We define three sequences \((A_n)\), \((G_n)\), and \((H_n)\) with the following rules:
- \(A_1\), \(G_1\), and \(H_1\) are the arithmetic mean, geometric mean, and harmonic mean of \(a\) and \(b\), respectively.
- For \(n \geq 2\), \(A_n\), \(G_n\), and \(H_n\) are the arithmetic mean, geometric mean, and harmonic mean of \(G_{n-1}\) and \(H_{n-1}\), respectively.

Determine the following statements:
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. \(H_1 < H_2 < H_3 < \dots\)

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