
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  $f^1(x)=x^3-3x$ . Let  $f^n(x)=f(f^{n-1}(x))$ . Let  $\mathcal{R}$  be the set of roots of  $\tfrac{f^{2022}(x)}{x}$ . If
\[\sum_{r\in\mathcal{R}}\frac{1}{r^2}=\frac{a^b-c}{d}\]
for positive integers  $a,b,c,d$ , where  $b$  is as large as possible and  $c$  and  $d$  are relatively prime, find  $a+b+c+d$ .

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