
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 quadratic function $f(x)= \frac {1}{3}x^{2}+ \frac {2}{3}x$. The sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and the point $(n,S_{n})$ (where $n\in\mathbb{N}^{*}$) is on the graph of the quadratic function $y=f(x)$.  
(Ⅰ) Find the general formula for the sequence $\{a_{n}\}$.  
(Ⅱ) Let $b_{n}=a_{n}a_{n+1}\cos [(n+1)\pi]$ (where $n\in\mathbb{N}^{*}$), and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$. If $T_{n}\geqslant tn^{2}$ holds for all $n\in\mathbb{N}^{*}$, find the range of values for the real number $t$.  
(Ⅲ) Determine whether there exist terms in the sequence $\{a_{n}\}$ such as $a_{n_{1}}$, $a_{n_{2}}$, $a_{n_{3}}$, …, $a_{n_{k}}$ that can form a geometric sequence $\{a_{n_{k}}\}$ with the first term $a_{1}$ and common ratio $q(0 < q < 5)$. If such terms exist, write the expression for $n_{k}$ in terms of $f(x)$; if not, explain why.

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