
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 two polygons $P_1$ and $P_2$, where $P_1$ is an equiangular decagon (10 sides) and each angle of $P_2$ is $kx$ degrees, with $k$ being an integer greater than $1$. If $\frac{kx}{x}$ determines the ratio between the number of sides of $P_2$ and $P_1$, find the smallest integer value of $k$ such that $P_2$ can exist as a valid polygon whose interior angles are also expressed in terms of $x$.

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