
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
$PQRS$ is a square and $X$ and $Y$ are the midpoints of $\overline{QR}$ and $\overline{RS},$ respectively.  Find $\cos \phi.$

[asy]
unitsize(1.5 cm);

draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((0,0)--(3,1.5));
draw((0,0)--(1.5,3));
label("$P$", (0,0), SW);
label("$Q$", (0,3), NW);
label("$R$", (3,3), NE);
label("$S$", (3,0), SE);
label("$X$", (1.5,3), N);
label("$Y$", (3,1.5), E);
label("$\phi$", (.8,.8), SW);
[/asy]

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