
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 \( a_1, a_2, \cdots, a_k \) be a finite arithmetic sequence. Given \( a_4 + a_7 + a_{10} = 17 \), \( a_4 + a_5 + a_6 + \cdots + a_{14} = 77 \), and \( a_k = 13 \), determine \( k \).

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