
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 that \( n \geq 5 \) is a positive integer, and \( a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} \geq 0 \) with the conditions

\[ 
\sum_{i=1}^{n} a_{i}^{2} = 1 \quad \text{and} \quad \sum_{i=1}^{n} b_{i} = 1 
\]

find the maximum value of 

\[
\sum_{i=1}^{n} a_{i}^{1+b_{i}}
\]

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