
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
The water storage volume of a reservoir changes over time, represented by $t$ (unit: month), with the beginning of the year as the starting point. According to historical data, the approximate functional relationship of the water storage volume $v$ (unit: billion cubic meters) concerning $t$ is $v(t)= \begin{cases} \frac {1}{240}(-t^2+15t-51)e^t+50(0 < t\leqslant 9) \\ 4(t-9)(3t-41)+50(9 < t\leqslant 12)\\ \end{cases}$.

1. If the water storage volume is less than 50, that period is considered a drought period. Representing the $i^{th}$ month as $i-1 < t\leqslant i$ $(i=1,2,…12)$, which months of the year are drought periods?
2. Find the maximum water storage volume of the reservoir within a year (assume $e^{3}=20$).

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