
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 a sequence $\left\{ a_n \right\}$ that satisfies for any $n\in N^{*}$, we have $a_1^3+a_2^3+\cdots +a_n^3=(a_1+a_2+\cdots +a_n)^2$, and $a_n > 0$.  
(1) Find the general formula for the sequence $\left\{ a_n \right\}$.  

(2) Let the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_n\cdot a_{n+2}} \right\}$ be $S_n$, and the inequality $S_n > \frac{1}{3}\log_a(1-a)$ holds for any positive integer $n$, find the range of the real number $a$.

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