
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
\( S \) is the intersection of the two lines \( l_{1}: 7x + 5y + 8 = 0 \) and \( l_{2}: 3x + 4y - 13 = 0 \). On the line passing through points \( P(3, 7) \) and \( Q(11, 13) \), there are two points \( A \) and \( B \) such that \( P \) lies between \( A \) and \( Q \), and \( B \) lies between \( P \) and \( Q \). Moreover, the ratios \( \left| \frac{PA}{AQ} \right| = \left| \frac{PB}{BQ} \right| = \frac{2}{3} \). Without finding the coordinates of \( S \), find the equations of the lines \( SA \) and \( SB \).

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