
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 function $f(x) = ax^2 + b\ln x$, with $a, b \in \mathbb{R}$, and given that $f(1)= \frac{1}{2}$ and $f'(2)=1$.  
(Ⅰ) Find the equation of the tangent line to the graph of $f(x)$ at the point $(1, f(1))$.  
(Ⅱ) Find the range of $f(x)$ on the interval $[1, \sqrt{e}]$.

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