
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
Let  $\{x\}$  denote the fractional part of  $x$ , which means the unique real  $0\leq\{x\}<1$  such that  $x-\{x\}$  is an integer. Let  $f_{a,b}(x)=\{x+a\}+2\{x+b\}$  and let its range be  $[m_{a,b},M_{a,b})$ . Find the minimum value of  $M_{a,b}$  as  $a$  and  $b$  range along all real numbers.

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