
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
$ABCD$ is a square and point $P$ and $Q$ are the midpoints of side $\overline{BC}$ and $\overline{CD}$, respectively. Line $AP$ and line $BQ$ intersect at point $X$. Calculate $\sin \theta$ where $\theta$ is the angle formed by lines $AP$ and $BQ$ at point $X$.

[asy]
unitsize(1.5 cm);

draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((0,0)--(4,2));
draw((0,4)--(2,0));
label("$A$", (0,0), SW);
label("$B$", (0,4), NW);
label("$C$", (4,4), NE);
label("$D$", (4,0), SE);
label("$Q$", (4,2), E);
label("$P$", (2,4), N);
label("$X$", (1.8, 1.2), N);  // Adjust according to actual intersection
label("$\theta$", (1,1), SW);
[/asy]

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