
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 $p$ and $q$ be positive real numbers with $p + q = 1$. We define three sequences $(A_n),$ $(G_n),$ and $(H_n)$ as follows. Initially, $A_1 = px + qy$, $G_1 = \sqrt{xy}$, and $H_1$ is the weighted harmonic mean of $x$ and $y$ given by $H_1 = \frac{1}{\frac{p}{x} + \frac{q}{y}}$. For $n \ge 2$, $A_n$ and $H_n$ are the weighted arithmetic mean and the weighted harmonic mean of $A_{n - 1}$ and $H_{n - 1}$ respectively, both with weights $p$ and $q$, and $G_n=\sqrt{A_{n-1}H_{n-1}}$.

Consider the following statements similar to the original problem, and determine which hold:
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. $H_1 < H_2 > H_3 < \dotsb.$

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