
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
1. The point \( C(13,11) \) is on the line segment joining \( A(1,2) \) and \( B(17,14) \). Point \( C \) has the property that the ratio of lengths \( AC: CB \) equals \( 3: 1 \) because
$$
\frac{13-1}{17-13}=\frac{12}{4}=\frac{3}{1} \text { and } \frac{11-2}{14-11}=\frac{9}{3}=\frac{3}{1} \text {. }
$$

In general, suppose that line segment \( AB \) is not horizontal or vertical, that \( C \) is on \( AB \), and that \( C \) is not at \( A \) or \( B \). The ratio of lengths \( AC: CB \) equals \( m: n \) exactly when the \( x \)-coordinate of \( C \) splits the \( x \)-coordinates of \( A \) and \( B \) in the ratio \( m: n \) and the \( y \)-coordinate of \( C \) splits the \( y \)-coordinates of \( A \) and \( B \) in the ratio \( m: n \).

(a) In the example above, determine the length of \( AC \) and the length of \( CB \).

(b) The point \( J(5,5) \) is on the line segment joining \( G(11,2) \) and \( H(1,7) \). Determine the ratio of the lengths \( GJ: JH \).

(c) Determine the coordinates of the point \( F \) on the line segment joining \( D(1,6) \) and \( E(7,9) \) so that the ratio of lengths \( DF: FE \) equals \( 1: 2 \).

(d) The point \( M(7,5) \) is on the line segment joining \( K(1, q) \) and \( L(p, 9) \) so that the ratio of lengths \( KM: ML \) equals \( 3: 4 \). Determine the values of \( p \) and \( q \).

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