
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]


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Given a real number $a > 0$, the function $f(x)=\sqrt{\frac{1-x^{2}}{1+x^{2}}}+a\sqrt{\frac{1+x^{2}}{1-x^{2}}}$.

$(1)$ When $a=1$, find the minimum value of $f(x)$;

$(2)$ When $a=1$, determine the monotonicity of $f(x)$ and explain the reason;

$(3)$ Find the range of the real number $a$ such that for any three real numbers $r$, $s$, $t$ in the interval $\left[ -\frac{2\sqrt{5}}{5},\frac{2\sqrt{5}}{5} \right]$, there exists a triangle with side lengths $f(r)$, $f(s)$, and $f(t)$.

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