
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
Consider a sequence of positive numbers $\{a_n\}$ (where $n$ is a positive integer), with $S_n$ being the sum of the first $n$ terms of $\{a_n\}$. If the point $(a_n, S_n)$ lies on the curve of the function $y=\frac {c^{2}-x}{c-1}$, where $c$ is a positive constant and $c \neq 1$:
1. Determine the general formula for the $\{a_n\}$ sequence.
2. Determine whether there exists a positive integer $M$ such that for all $n > M$, the product $a_1 \cdot a_3 \cdot a_5 \cdots a_{2n-1} > a_{101}$ holds true consistently. If it does exist, find the range of values for $c$ and the minimum value for $M$.
3. Suppose there is an arithmetic sequence $\{b_n\}$ such that for any positive integer $n$, the following is true: $b_1a_n+b_2a_{n-1}+b_3a_{n-2}+\cdots+b_{n-1}a_2+b_na_1 =3^{n}- \frac {5}{3}n-1$. Find the general formula for the $\{b_n\}$ sequence and the value of $c$.

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