
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 equations:

1. \(\sin \left( \frac{4 \pi (2k + 1)}{7} \right) = 0 \rightarrow 7m = 8k + 4, m \in \mathbb{Z} \rightarrow \left\{
\begin{array}{c}
k = -4 + 7t \\
m = -4 + 8t, t \in \mathbb{Z}
\end{array}
\right.\)

2. \(\sin \left( \frac{4 \pi (3k - 2)}{7} \right) = 0 \rightarrow 7n = 12k - 8, m \in \mathbb{Z} \rightarrow \left\{
\begin{array}{c}
k = 3 + 7s \\
m = 4 + 12s, s \in \mathbb{Z}
\end{array}
\right.\)

Finally, the equation \(7t - 4 = 7s + 3 \rightarrow t = s + 1\). Then \(k = 3 + 7s\).

Translate and solve for \(k\).

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