
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]


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Given that in triangle \( \triangle ABC \), the coordinates of the vertices are \( A\left(x_{1}, y_{1}\right) \), \( B\left(x_{2}, y_{2}\right) \), and \( C\left(x_{3}, y_{3}\right) \). Points \( E \) and \( F \) lie on lines \( AC \) and \( AB \) respectively, and \(\frac{\overline{AF}}{\overline{FB}} = \frac{m}{l}\), \(\frac{\overline{AE}}{\overline{EC}} = \frac{n}{l}\) (\(m \neq -l\), \(n \neq -l\), and \( l + m + n \neq 0 \)). Find the coordinates of the intersection point of lines \( BE \) and \( CF \).

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