
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 an equilateral triangle $\triangle_{1}$ with an area of $S_{1}$, an inscribed circle is drawn, and then another equilateral triangle $\triangle_{2}$ is drawn inscribed within that circle with an area of $S_{2}$. This process is repeated to create a sequence of equilateral triangles $\triangle_{3}, \triangle_{4}, \cdots$, with corresponding areas $S_{3}, S_{4}, \cdots$. Given that $S_{1} = 1$, and $T_{n} = S_{1} + S_{2} + \cdots + S_{n}$, find the value of $\lim_ {n \rightarrow + \infty} T_{n}$.

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