
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 a common difference $d \neq 0$, $S_7=35$, and $\{a_2\}$, $\{a_5\}$, $\{a_{11}\}$ form a geometric sequence.

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

$(2)$ If $T_n$ is the sum of the first $n$ terms of the sequence $\left\{\frac{1}{a_na_{n+1}}\right\}$, and there exists $n \in \mathbb{N}^*$ such that $T_n - \lambda a_{n+1} \geqslant 0$ holds, find the range of values for the real number $\lambda$.

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