
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 functions $f\left( x \right)=-{{4}^{x}}+{{2}^{x+1}}-1$ and $g\left( x \right)=1g\left( a{{x}^{2}}-4x+1 \right)$, determine the range of values for the real number $a$ such that for any ${{x}_{1}}\in R$, there exists ${{x}_{2}}\in R$ such that $f\left( {{x}_{1}} \right)=g\left( {{x}_{2}} \right)$.

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