
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
 Let  $a$  and  $b$  be two-digit positive integers. Find the greatest possible value of  $a+b$ , given that the greatest common factor of  $a$  and  $b$  is  $6$ .

*Proposed by Jacob Xu*

<details><summary>Solution</summary>*Solution*.  $\boxed{186}$ We can write our two numbers as  $6x$  and  $6y$ . Notice that  $x$  and  $y$  must be relatively prime. Since  $6x$  and  $6y$  are two digit numbers, we just need to check values of  $x$  and  $y$  from  $2$  through  $16$  such that  $x$  and  $y$  are relatively prime. We maximize the sum when  $x = 15$  and  $y = 16$ , since consecutive numbers are always relatively prime. So the sum is  $6 \cdot (15+16) = \boxed{186}$ .</details>

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