
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. Define four sequences $(A_n),$ $(G_n),$ $(H_n)$, and $(Q_n)$, where $A_1$, $G_1$, $H_1$, and $Q_1$ are the arithmetic mean, geometric mean, harmonic mean, and quadratic mean of $x$ and $y$, respectively. For $n \ge 2$, define $A_n$, $G_n$, $H_n$, $Q_n$ as the arithmetic mean, geometric mean, harmonic mean, and quadratic mean of $A_{n-1}$ and $H_{n-1}$, respectively.

Determine the behavior of the sequences $A_n$, $G_n$, $H_n$, and $Q_n$ as $n$ increases.

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