
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 $f(x)=1- \sqrt{x^2+1}$ and $g(x)=\ln (ax^2-2x+1)$. If for any $x_1 \in \mathbb{R}$, there exists a real number $x_2$ such that $f(x_1)=g(x_2)$ holds, then find the range of the real number $a$.

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