
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 the positive integers  $n$  that are not divisible by  $3$  if the number  $2^{n^2-10}+2133$  is a perfect cube.

<details><summary>Note</summary><span style="color:#BF0000">The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation  $x^3=2^{n^2-10}+2133$  where  $x,n \in \mathbb{N}$  and  $3\nmid n$ .</span></details>

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