
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 the equation $(x^{2}-2x+m)(x^{2}-2x+n)=0$ with its four roots forming an arithmetic progression with a first term of $\frac{1}{4}$. Let acute triangle $ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, where $b=4|m-n|$ and $A=2B$. Find the range of values for $a$.

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