
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 let $P_1$ be the power mean of order 2 (quadratic mean) of $x$ and $y$. Define three sequences $(A_n), (G_n), (H_n)$ as follows: $A_1, G_1, H_1$ are the arithmetic mean, geometric mean, and harmonic mean of $x$ and $y$, respectively. Then for $n \geq 2$, $A_n, G_n, H_n$ are the arithmetic mean, geometric mean, harmonic mean of $P_{n-1}$ and $H_{n-1}$, respectively. Also, define $P_n$ for $n \geq 2$ as the power mean of order 2 of $P_{n-1}$ and $H_{n-1}$. Determine whether the following statements are true:
1. $A_1 > A_2 > A_3 > \dots$.
2. $G_1 > G_2 > G_3 > \dots$.
3. $H_1 > H_2 > H_3 > \dots$.
4. $P_1 > P_2 > P_3 > \dots$.

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