
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
Let \( \triangle ABC \) be a triangle with \( AB = 5 \), \( BC = 8 \), and \( CA = 7 \). Let \( \Gamma \) be a circle internally tangent to the circumcircle of \( \triangle ABC \) at \( A \) and also tangent to segment \( BC \). \( \Gamma \) intersects \( AB \) and \( AC \) at points \( D \) and \( E \), respectively. Determine the length of segment \( DE \).

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