
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 that $a > 0$ is a real number, the function $f(x)= \sqrt { \dfrac {1-x^{2}}{1+x^{2}}}+a \sqrt { \dfrac {1+x^{2}}{1-x^{2}}}$ is provided.
(1) When $a=1$, find the minimum value of $f(x)$;
(2) When $a=1$, determine the monotonicity of $f(x)$ and explain the reasoning;
(3) Find the range of real number $a$ such that for any three real numbers $r$, $s$, $t$ in the interval $\left[- \dfrac {2 \sqrt {5}}{5}, \dfrac {2 \sqrt {5}}{5}\right]$, 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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