
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
In a certain river channel, an overgrowth of a type of algae has occurred. The environmental protection department decided to introduce a biological purification agent to clean the water body. Due to technical reasons, the path length of the purification agent released in the $t$th minute is $p=140-|t-40|$ (unit: $m$), and the width of the water body purified by the purification agent $q$ (unit: $m$) is a function of time $t$ (unit: minute): $q(t)=1+a^{2}t$ ($a$ is determined by the amount of purification agent released per unit time, assume $a$ is a constant, and $a\in N^{*}$).  
(1) Try to write the expression for the area $S(t)$ of the water body purified in the $t$th minute $(1\leqslant t\leqslant 60,t\in N^{*})$;  
(2) Find the minimum value of $S(t)$.

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