
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
Let $m$ be the number of coins in the treasure. By morning, the number of remaining coins $N$ is given by

$$
\begin{gathered}
N = \frac{2}{3}\left(\frac{2}{3}\left(\frac{2}{3}(m-1) - 1\right) - 1\right) = \frac{2^3}{3^3}(m-1) - \frac{2}{3} - \frac{2^2}{3^2} \\
N = \frac{2^3}{3^3}(m-1) - \frac{10}{9} = 3n \\
8(m-1) - 30 = 81k, \text{ hence }\left\{\begin{array}{c}
m-1 = 3t, \\
8t - 27k = 10
\end{array}\right., \left\{\begin{array}{c}
m = 3t + 1 \\
k = 2s \\
4t - 27s = 5
\end{array}\right.
\right.
\left\{\begin{array}{c}
m = 3t + 1 \\
k = 2s, \\
t = 8 + 27u, \\
s = 1 + 4u
\end{array}\right.
\end{gathered}
$$

Given the constraints

$$
\begin{gathered}
110 \leq 81u + 25 \leq 200, \text{ thus } u=2, m=187. \\
m_1 = 1 + \frac{187 - 1}{3} = 63, \quad \widetilde{m}_1 = 63 + 18 = 81 \\
- \text{ coins of the first pirate; } \\
m_2 = 1 + \frac{187 - 63 - 1}{3} = 42, \quad \widetilde{m}_2 = 42 + 18 = 60 \\
m_3 = 1 + \frac{187 - 63 - 42 - 1}{3} = 28, \quad \widetilde{m}_3 = 28 + 18 = 46 \\
- \text{ coins of the third pirate; } \\
m_0 = \frac{187 - 63 - 42 - 28}{3} = 18 \\
\text{- coins each receives in the morning. }
\end{gathered}
$$

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