
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
Starting from midnight, when the minute hand and the hour hand coincide for the $i$th time $(i\in N^{*})$ on the clock face, the minute hand has moved $a_{i}$ minutes, calculate the sum of all such $a_{i}$ within 24 hours (including the 24th hour).

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