
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 \(\{a_n\}\) which satisfies the relation: \(a_1 = 1\) and \(a_{n+1} = a_n + a_n^2\) for \(n \in \mathbb{N}^*\). Let
\[
S_n = \frac{1}{(1+a_1)(1+a_2)\cdots(1+a_n)}, \quad T_n = \sum_{k=1}^n \frac{1}{1+a_k}
\]
Find the value of \(S_n + T_n\).

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