
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
The vertices of triangle \( \triangle ABC \) are \( A(0,0), B(0,420), C(560,0) \). A die has its six faces marked with \( A, A, B, B, C, C \). A point \(P_{1}=(k, m)\) is chosen inside \( \triangle ABC \), and subsequent points \( P_{2}, P_{3}, P_{4}, \cdots \) are generated according to the following rule: If \( P_{n} \) is already chosen, roll the die to get a label \( L, L \in \{A, B, C\} \). Then \( P_{n+1} \) is the midpoint of \( P_{n} \) and \( L \). Given \( P_{7}=(14,92) \), find the value of \( k+m \).

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