
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 triples  $(k, m, n)$  of positive integers such that  $m$  is a prime and:

(1)  $kn$  is a perfect square;

(2)  $\frac{k(k-1)}{2}+n$  is a fourth power of a prime;

(3)  $k-m^2=p$  where  $p$  is a prime;

(4)  $\frac{n+2}{m^2}=p^4$ . 

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