
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 sequence $\{a_n\}$ is an arithmetic sequence, and $S_n$ is the sum of its first $n$ terms, satisfying $S_2=4$, $S_5=25$. The sequence $\{b_n\}$ satisfies $b_n= \frac {1}{a_{n}-a_{n+1}}$, and $T_n$ is the sum of the first $n$ terms of sequence $\{b_n\}$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) If for any $n\in\mathbb{N}^*$, the inequality $\lambda T_n < n+8\cdot(-1)^n$ always holds, find the range of the real number $\lambda$.  
(3) Does there exist positive integers $m$, $n$ ($1<m<n$) such that $T_1$, $T_m$, $T_n$ form a geometric sequence? If so, find all values of $m$ and $n$; if not, explain why.

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