
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
Find the greatest possible real value of  $S$  and smallest possible value of  $T$  such that for every triangle with sides  $a,b,c$   $(a\le b\le c)$  to be true the inequalities:  $$ S\le\frac{(a+b+c)^2}{bc}\le T. $$  

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