
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
For any positive integer  $a$ , define  $M(a)$  to be the number of positive integers  $b$  for which  $a+b$  divides  $ab$ . Find all integer(s)  $a$  with  $1\le a\le 2013$  such that  $M(a)$  attains the largest possible value in the range of  $a$ .

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