
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
1. Given the curve $y=x+\ln x$, find the general equation of the tangent line at the point $(1,1)$.
2. Find the interval where the function $f(x)=x^2-2\ln x$ is monotonically decreasing.
3. Given the integer pair table below:
   $(1,1)$
   $(1,2)$, $(2,1)$
   $(1,3)$, $(2,2)$, $(3,1)$
   $(1,4)$, $(2,3)$, $(3,2)$, $(4,1)$
   $(1,5)$, $(2,4)$, $(3,3)$, $(4,2)$, $(5,1)$
   ... ...
   What is the 10th pair in the 20th row from left to right?
4. If $f(x)=-\frac{1}{2}x^2+b\ln x$ is a decreasing function on $(1,+\infty)$, find the range of $b$.

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