
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
Square $PQRS$ has sides of length 1. Points $M$ and $N$ are on $\overline{QR}$ and $\overline{RS},$ respectively, so that $\triangle PMN$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\overline{PM}.$ The length of a side of this smaller square is $\frac{d-\sqrt{e}}{f},$ where $d, e,$ and $f$ are positive integers and $e$ is not divisible by the square of any prime. Find $d+e+f.$

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