
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
Say a real number  $r$  is \emph{repetitive} if there exist two distinct complex numbers  $z_1,z_2$  with  $|z_1|=|z_2|=1$  and  $\{z_1,z_2\}\neq\{-i,i\}$  such that 
\[
z_1(z_1^3+z_1^2+rz_1+1)=z_2(z_2^3+z_2^2+rz_2+1).
\]
There exist real numbers  $a,b$  such that a real number  $r$  is \emph{repetitive} if and only if  $a < r\le b$ . If the value of  $|a|+|b|$  can be expressed in the form  $\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ , find  $100p+q$ .

*Proposed by James Lin*

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