
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 $f(x)=\ln (x^{2}+1)$, $g(x)= \frac {1}{x^{2}-1}+a$.  
$(1)$ If one of the extreme points of $f(x)$ is at a distance of $1$ from the line $l$: $2 \sqrt {2}x+y+a+5=0$, find the value of $a$;  
$(2)$ Find the number of roots of the equation $f(x)=g(x)$.

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