
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
Find all primes  $ p$  for that there is an integer  $ n$  such that there are no integers  $ x,y$  with  $ x^3 \plus{} y^3 \equiv n \mod p$  (so not all residues are the sum of two cubes).

E.g. for  $ p \equal{} 7$ , one could set  $ n \equal{} \pm 3$  since  $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$ , thus  $ x^3 \plus{} y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7$  only.

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