
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
A sequence \( t_{1}, t_{2}, t_{3}, \ldots \) is defined by

\[ t_{n} = \begin{cases}
\frac{1}{7^{n}} & \text{when } n \text{ is odd} \\
\frac{2}{7^{n}} & \text{when } n \text{ is even}
\end{cases} \]

for each positive integer \( n \). Determine the sum of all of the terms in this sequence, i.e., calculate \( t_{1} + t_{2} + t_{3} + \cdots \).

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