
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
Consider a triangle \( \triangle ABC \) with base angles \( A \) and \( B \) such that \( B > A \). An external angle at vertex \( B \) is \( B_{ext} \). The altitude from vertex \( C \) to base \( AB \) divides angle \( C \) into two parts, \( C_1 \) and \( C_2 \), with \( C_2 \) adjacent to side \( a \) (opposite angle \( A \)). Determine the relationship between the angles \( C_1 \), \( C_2 \), \( A \), \( B_{ext} \).
A) \( C_1 + C_2 = A + B \)
B) \( C_1 - C_2 = B - A \)
C) \( C_1 + C_2 = A + B_{ext} - 180^\circ \)
D) \( C_1 - C_2 = A + B_{ext} - 180^\circ \)

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