
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
The sum of the first 4 terms of an arithmetic sequence $\{a_n\}$, which has a common difference not equal to $0$, is $S_4=10$, and $a_1$, $a_8$, $a_{29}$ form a geometric sequence.

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

$(2)$ Let $T_n$ be the sum of the first $n$ terms of the sequence $\left\{\frac{1}{{a_n}{a_{n+1}}}\right\}$. If $7T_n \leqslant \lambda a_{n}+1$ holds for $\forall n\in\mathbb{N}^*$, find the minimum value of the real number $\lambda$.

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