
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
Fix a point \( O \) on the plane. Any other point \( A \) is uniquely determined by its radius vector \(\overrightarrow{OA}\), which we will denote simply as \(\vec{A}\). A fixed point \( O \) is called the initial point. We will also introduce the symbol \((ABC)\) - the simple ratio of three points \( A, B, C \) that lie on a straight line; points \( A \) and \( B \) are called the base points, and point \( C \) is the dividing point. The symbol \( (ABC) \) is a number defined by the formula:

$$
\overrightarrow{CA} = (ABC) \cdot \overrightarrow{CB}
$$

For two collinear vectors \(\vec{a}\) and \(\vec{b}\), a unique number \(\lambda\) is defined such that \(\vec{a} = \lambda \overrightarrow{b}\). We introduce the notation \(\lambda = \frac{\vec{a}}{\vec{b}}\) (the ratio of collinear vectors). In this case, we can write:

$$
(ABC) = \frac{\overrightarrow{CA}}{\overrightarrow{CB}}
$$

Derive a formula that expresses the vector \(\vec{C}\) in terms of the vectors \(\vec{A}\) and \(\vec{B}\), given that \((ABC) = k\).

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