
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 $f(x)=\ln x- \frac{1}{4}x+ \frac{3}{4x}-1$ and $g(x)=x^2-2bx+4$. For any $x_1 \in (0,2)$, there exists $x_2 \in [1,2]$ such that $f(x_1) \geqslant g(x_2)$. Determine the range of the real number $b$.

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