
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 sequence \( \left\{a_{n}\right\} \) satisfies \( a_{1}=\frac{1}{2} \) and \( a_{n+1}=a_{n}^{2}+a_{n} \). Define \( b_{n}=\frac{1}{1+a_{n}} \), \( S_{n}=b_{1}+b_{2}+\cdots+b_{n} \), and \( P_{n}=b_{1} b_{2} \cdots b_{n} \). Find the value of \( 2 P_{n}+S_{n} \).

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