
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 \( r_k \) denote the remainder when \( \binom{127}{k} \) is divided by 8. Compute \( r_1 + 2 r_2 + 3 r_3 + \cdots + 63 r_{63} \).

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