
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 $f\left(x\right)=x\ln x+ax+b$ and the tangent line at $\left(1,f\left(1\right)\right)$ is $2x-2y-1=0$.  
$(1)$ Find the values of real numbers $a$ and $b$.  
$(2)$ Determine the monotonic intervals of $f\left(x\right)$.

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