
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
For a geometric sequence $g_1,$ $g_2,$ $g_3,$ $\dots,$ let
\[S_n = g_1 + g_2 + g_3 + \dots + g_n,\]and let
\[T_n = S_1 + S_2 + S_3 + \dots + S_n.\]If $S_{2019}$ is known, for which smallest integer $n$ can you determine $T_n$ uniquely?

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