
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 right $\triangle DEF$, the sides are $DE=5$, $EF=12$, and $DF=13$. A square $PQRS$ is inscribed in $\triangle DEF$ such that $P$ and $Q$ lie on $\overline{DF}$, $R$ on $\overline{DE}$, and $S$ on $\overline{EF}$. Additionally, a circle is inscribed inside the square $PQRS$. What is the side length of the square and the radius of the circle?

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