
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
Consider a diagram with 8x5 lattice grid where each point is one unit apart from its nearest neighbors. Segment $PQ$ meets segment $RS$ at point $T$. Point $P$ is located at $(0,4)$, point $Q$ at $(7,0)$, point $R$ at $(3,0)$, and point $S$ at $(5,3)$.

[asy]
unitsize(0.8cm);
for (int i=0; i<8; ++i) {
for (int j=0; j<5; ++j) {
dot((i,j));
};}
label("$P$",(0,4),W);
label("$Q$",(7,0),E);
label("$S$",(5,3),N);
label("$T$",(4,1.5),S);
dot((4,1.5));
label("$R$",(3,0),S);
draw((0,4)--(7,0),linewidth(0.7));
draw((3,0)--(5,3),linewidth(0.7));
[/asy]

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