
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 fixed triangles $ABC$ and $X'Y'Z'$ in the plane, we need to construct a triangle $XYZ$ such that its sides are parallel to the sides of triangle $X'Y'Z'$. The vertices $X, Y, Z$ of the triangle $XYZ$ should lie on the sides of triangle $ABC$ and not on their extensions. 

What are the conditions for the existence of such a triangle $XYZ$ and how many possible solutions are there? Additionally, how many solutions exist if the vertices of the desired triangle can also lie on the extensions of the sides of triangle $ABC$?

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