
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 $f(x) = a\ln x + bx$, where $a, b \in \mathbb{R}$.  
$(Ⅰ)$ When $a=1$ and $b=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$.  
$(Ⅱ)$ When $a > 0$ and $b=-2$, find the maximum value of $f(x)$ on the interval $[1,2]$.  
$(Ⅲ)$ When $a=1$, let $g(x) = f(x) + \sin x$. Determine if $g(x)$ has extreme values for $x \in (0, \pi]$. If it does, specify whether it is a maximum or minimum; if not, explain why.

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