
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
The bivariate functions  $f_0, f_1, f_2, f_3, \dots$  are sequentially defined by the relations  $f_0(x,y) = 0$  and  $f_{n+1}(x,y) = \bigl|x+|y+f_n(x,y)|\bigr|$  for all integers  $n \geq 0$ . For independently and randomly selected values  $x_0, y_0 \in [-2, 2]$ , let  $p_n$  be the probability that  $f_n(x_0, y_0) < 1$ . Let  $a,b,c,$  and  $d$  be positive integers such that the limit of the sequence  $p_1,p_3,p_5,p_7,\dots$  is  $\frac{\pi^2+a}{b}$  and the limit of the sequence  $p_0,p_2,p_4,p_6,p_8, \dots$  is  $\frac{\pi^2+c}{d}$ . Compute  $1000a+100b+10c+d$ .

*Proposed by Sean Li*

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