
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 an arithmetic sequence $\{a_n\}$ with the first term $a_1=1$, and the common difference $d$ being an integer, it satisfies $a_1+3 < a_3$ and $a_2+5 > a_4$. The sequence $\{b_n\}$ is defined by $b_{n}= \frac {1}{a_{n}\cdot a_{n+1}}$. Let $S_n$ be the sum of the first $n$ terms of $\{b_n\}$.  
(1) Find the general term $a_n$ of the sequence $\{a_n\}$.  
(2) If $S_2$ is the geometric mean of $S_1$ and $S_m$ (where $m \in \mathbb{N}^+$), find the value of $m$.

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