
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
Given integers \( a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7} \) satisfy \( \frac{5}{7}=\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\frac{a_{4}}{4!}+\frac{a_{5}}{5!}+\frac{a_{6}}{6!}+\frac{a_{7}}{7!} \), for \( i = 2, 3, \cdots, 7 \), such that \( 0 \leqslant a_{i} < i \), calculate the value of \( a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7} \).

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