
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
Call an odd prime  $p$  *adjective* if there exists an infinite sequence  $a_0,a_1,a_2,\ldots$  of positive integers such that \[a_0\equiv1+\frac{1}{a_1}\equiv1+\frac{1}{1+\frac{1}{a_2}}\equiv1+\frac{1}{1+\frac{1}{1+\frac{1}{a_3}}}\equiv\ldots\pmod p.\] What is the sum of the first three odd primes that are *not* adjective?

Note: For two common fractions  $\frac{a}{b}$  and  $\frac{c}{d}$ , we say that  $\frac{a}{b}\equiv\frac{c}{d}\pmod p$  if  $p$  divides  $ad-bc$  and  $p$  does not divide  $bd$ .

*2019 CCA Math Bonanza Individual Round #14*

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