
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
Pedrito's lucky number is  $34117$ . His friend Ramanujan points out that  $34117 = 166^2 + 81^2 = 159^2 + 94^2$  and  $166-159 = 7$ ,  $94- 81 = 13$ . Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of positive integers, and the difference of the first integers that occur in that sum is  $7$  and in the difference between the seconds it gives  $13$ . Which is the least lucky number that Pedrito can find? Find a way to generate all the positive integers with the properties mentioned above.

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