
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 $EFGH$ has sides of length 5, and $N$ is the midpoint of $\overline{GH}$. A circle with radius 2.5 and center $N$ intersects a circle with radius 5 and center $E$ at points $Q$ and $H$. What is the distance from $Q$ to $\overline{EG}$?

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