
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 $f(x)=x+\dfrac{a}{x}+a^{2}-2$ ($a\in \mathbb{R}$).

$(1)$ If $f(x)$ is an odd function, and it is increasing on the interval $(0,+\infty)$, find the value of $a$;

$(2)$ If the equation $|\log_{8}(x+1)|-a^{2}+f(1)=0$ has two different real roots $m$, $n$ in the interval $(-1,1)$, find the range of $a$, and calculate the value of $\dfrac{1}{m}+\dfrac{1}{n}$.

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