
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
Let the lengths of the sides of $\triangle A_nB_nC_n$ be $a_n$, $b_n$, $c_n$, for $n=1,2,3,\ldots$. If $b_1 > c_1$, $b_1+c_1=2a_1$, $a_{n+1}=a_n$, $b_{n+1}=\frac{c_n+a_n}{2}$, $c_{n+1}=\frac{b_n+a_n}{2}$, then the maximum value of $\angle A_n$ is \_\_\_\_\_\_.

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