
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 *compatible* if  $m \ge \tfrac{1}{2} n + 7$  and  $n \ge \tfrac{1}{2} m + 7$ .  A positive integer  $k \ge 1$  is called *lonely* if  $(k,\ell)$  is not compatible for any integer  $\ell \ge 1$ .  Find the sum of all lonely integers.

*Proposed by Evan Chen*

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