
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  $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$   with   $a+b+c=1 \}$  and   $f: M\to  R$  given as    $$ f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc} $$  Find the best (real) bounds  $\alpha$  and  $\beta$  such that  $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$  and determine whether any of them is achievable.

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