
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 two coprime positive integers  $p, q$ , define  $f(i)$  to be the remainder of  $p\cdot i$  divided by  $q$  for  $i = 1, 2,\ldots,q -1$ . The number  $i$  is called a**large**number (resp. **small** number) when  $f(i)$  is the maximum (resp. the minimum) among the numbers  $f(1), f(2),\ldots,f(i)$ . Note that  $1$  is both large and small. Let  $a, b$  be two fixed positive integers. Given that there are exactly  $a$  large numbers and  $b$  small numbers among  $1, 2,\ldots , q - 1$ , find the least possible number for  $q$ .
*Proposed by usjl*

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