
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
Suppose the real numbers $a$, $b$, and $c$ satisfy the system of inequalities $\begin{cases} & b+c\leqslant 4a \\ & c-b\geqslant 0 \\ & b\geqslant a > 0 \\ \end{cases}$, but do not satisfy $(a+b)^{2}+(a+c)^{2}=(ar)^{2}\left(r > 0\right)$. Determine the range of possible values for $r$.

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