
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 $f(x)=1- \sqrt {x+1}$ and $g(x)=\ln (ax^{2}-3x+1)$, if for any $x_1\in(0,+\infty)$, there exists an $x_2\in\mathbb{R}$ such that $f(x_1)=g(x_2)$ holds true, determine the maximum value of the real number $a$.

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