
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
Someone claims that the equation

$$
x_{1}+x_{2}+x_{3}=n
$$

where $n$ is a natural number, has solutions in natural numbers satisfying the requirement

$$
0<x_{1}<x_{2}<x_{3}
$$

In the cases where

$$
n=21, \quad 57 \quad \text { and } \quad 165
$$

there are exactly

$$
27\left(=3^{3}\right), \quad 243\left(=3^{5}\right), \quad 2187\left(=3^{7}\right)
$$

solutions, respectively. Is this a coincidence, or are we dealing with special cases of a general statement?

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