
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
Let $f(x)=-4^{x}+2^{x+1}-1$, $g(x)=\log_{10} (ax^{2}-4x+1)$. If for any $x_{1}\in \mathbb{R}$, there exists $x_{2}\in \mathbb{R}$ such that $f(x_{1})=g(x_{2})$, then the range of real number $a$ is ______.

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