
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 the sequence $\left\{a_{n}\right\}$ defined by $a_{1}=\frac{2}{3}$ and $a_{n+1}=a_{n}^{2}+a_{n+1}^{2}+\ldots+a_{1}^{2}$ for $n \in \mathbb{N}^{*}$, if $\frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\ldots+\frac{1}{a_{n}+1}<M$ holds for any $n \in \mathbb{N}^{*}$, find the minimum value of $M$.

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