
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
A pair of positive integers  $m, n$  is called *guerrera*, if there exists positive integers  $a, b, c, d$  such that  $m=ab$ ,  $n=cd$  and  $a+b=c+d$ . For example the pair  $8, 9$  is *guerrera* cause  $8= 4 \cdot 2$ ,  $9= 3 \cdot 3$  and  $4+2=3+3$ . We paint the positive integers if the following order:

We start painting the numbers  $3$  and  $5$ . If a positive integer  $x$  is not painted and a positive  $y$  is painted such that the pair  $x, y$  is *guerrera*, we paint  $x$ . 

Find all positive integers  $x$  that can be painted.

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