
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 \( f(x) = x^2 - r_2 x + r_3 \) for all real numbers \( x \), where \( r_2 \) and \( r_3 \) are some real numbers. Define a sequence \( \{ g_n \} \) for all nonnegative integers \( n \) by \( g_0 = 0 \) and \( g_{n+1} = f(g_n) \). Assume that \( \{ g_n \} \) satisfies the following three conditions:
1. \( g_{2i} < g_{2i+1} \) and \( g_{2i+1} > g_{2i+2} \) for all \( 0 \leq i \leq 2011 \).
2. There exists a positive integer \( j \) such that \( g_{i+1} > g_i \) for all \( i > j \).
3. \( \{ g_n \} \) is unbounded.

If \( A \) is the greatest number such that \( A \leq |r_2| \) for any function \( f \) satisfying these properties, find \( A \).

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