
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 quadratic function $f\left( x \right)=a{{x}^{2}}+bx+c$, where its derivative is ${f}'\left( x \right)$, the inequality $f\left( x \right)\geqslant {f}'\left( x \right)$ holds true for all $x\in R$. Determine the maximum value of $\dfrac{{{b}^{2}}}{{{a}^{2}}+2{{c}^{2}}}$.

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