
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 distinct terms, the sum of the first four terms is $S_4=14$, and $a_1, a_3, a_7$ form a geometric sequence.

(1) Find the general term formula for the sequence $\{a_n\}$.

(2) Let $T_n$ denote the sum of the first $n$ terms of the sequence $\{\frac{1}{a_n a_{n+1}}\}$. If there exists a real number $\lambda$ such that $\lambda T_n \leq a_{n+1}$ holds for all positive integers $n$, find the maximum value of $\lambda$.

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