
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
An astronomer discovered that the intervals between the appearances of comet $2011 Y$ near the planet $12 I V 1961$ are successive terms of a decreasing geometric progression. The three latest intervals (in years) are the roots of the cubic equation \( t^{3}-c t^{2}+350 t-1000=0 \), where \( c \) is a constant. What will be the duration of the next interval before the comet's appearance?

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