
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
In $\triangle XYZ, XY = 9, YZ = 8, ZX = 7$, and side $ZX$ is extended to a point $Q$ such that $\triangle QXY$ is similar to $\triangle QYZ$. Find the length of $QY$.
[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair X=origin, Q=(1.5,5), Y=(9,0), Z=Q+2.5*dir(Q--Y); draw(X--Q--Z--X--Y--Z); label("X", X, W); label("Y", Y, E); label("Z", Z, NE); label("Q", Q, NW); label("9", (4.5,0), S); label("8", Y + 2.5*dir(Y--Z), NE); label("7", 3.5*dir(X--Z), SE); [/asy]

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