
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
If a three-digit positive integer $M$ can be expressed in the form of $m(m+3)$, where $m$ is a positive integer, then $M$ is called a "lucky number". For example, for the three-digit number $270$, since $270=15\times (15+3)$, therefore $270$ is a "lucky number"; another example is the three-digit number $102$, since $102=1\times 102=2\times 51=3\times 34=6\times 17$, therefore $102$ is not a "lucky number". According to the question, the largest "lucky number" is ______; if $M$ and $N$ are both "lucky numbers", and $M-N=350$, then the sum of all satisfying values of $N$ is ______.

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