
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 the point $(1, \frac{1}{3})$ is on the graph of the function $f(x)=a^{x}(a > 0$ and $a\neq 1)$, and the sum of the first $n$ terms of the geometric sequence ${a_{n}}$ is $f(n)-c$, the first term of the sequence ${b_{n}}(b_{n} > 0)$ is $c$, and the sum of the first $n$ terms $S_{n}$ satisfies $S_{n}-S_{n-1}= \sqrt {S_{n}}+ \sqrt {S_{n-1}}(n\geqslant 2)$. Let $T_{n}$ be the sum of the first $n$ terms of the sequence ${\frac {1}{b_{n}b_{n+1}}}$,
$(1)$ Find the general terms of the sequences ${a_{n}}$ and ${b_{n}}$;
$(2)$ If for any positive integer $n$, the inequality $t^{2}-2mt+ \frac {1}{2} > T_{n}$ always holds when $m\in[-1,1]$, find the range of values of the real number $t$;
$(3)$ Does there exist positive integers $m$ and $n$, with $1 < m < n$, such that $T_{1}$, $T_{m}$, $T_{n}$ form a geometric sequence? If so, find the values of $m$ and $n$; if not, explain the reason.

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