
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
Given a triangle with side lengths  $a,\ b,\ c$ . Let  $a,\ b,\ c$  vary, find the range of  $\frac{a^2+b^2+c^2}{ab+bc+ca}$ .

*2010 Tokyo Institute of Technology Admission Office entrance exam, Problem I-2/Science*

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