
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
Martin has two boxes  $A$  and  $B$ . In the box  $A$  there are  $100$  red balls numbered from  $1$  to  $100$ , each one with one of these numbers. In the box  $B$  there are  $100$  blue balls numbered from  $101$  to  $200$ , each one with one of these numbers. Martin chooses two positive integers  $a$  and  $b$ , both less than or equal to  $100$ , and then he takes out  $a$  balls from box  $A$  and  $b$  balls from box  $B$ , without replacement. Martin's goal is to have two red balls and one blue ball among all balls taken such that the sum of the numbers of two red balls equals the number of the blue ball.
	
	What is the least possible value of  $a+b$  so that Martin achieves his goal for sure? For such a minimum value of  $a+b$ , give an example of  $a$  and  $b$  satisfying the goal and explain why every  $a$  and  $b$  with smaller sum cannot accomplish the aim.

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