
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 positive integers  $1 \leq k \leq 6$  such that for any prime  $p$ , satisfying  $p^2=a^2+kb^2$  for some positive integers  $a, b$ , there exist positive integers  $x, y$ , satisfying  $p=x^2+ky^2$ .

<details><summary>Remark on 10.4</summary>It also appears as ARO 2010 10.4 with the grid changed to  $10 \times 10$  and  $17$  changed to  $5$ , so it will not be posted.</details>

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