
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 the prediction that the number of shared bikes deployed and lost in a certain area in the $n^{th}$ month ($n \in \mathbb{N}^*$) are $a_n$ and $b_n$ (in units of bikes) respectively, where $a_n = \begin{cases} 5n^4 + 15, & 1 \leqslant n \leqslant 3 \\ -10n + 470, & n \geqslant 4 \end{cases}$ and $b_n = n + 5$. The remaining number of shared bikes at the end of the $n^{th}$ month is the difference between the cumulative deployment and cumulative loss up to that month.
(1) Calculate the remaining number of shared bikes in this area at the end of the $4^{th}$ month.
(2) It is known that the capacity of the shared bike parking spots in this area in the $n^{th}$ month is $S_n = -4(n - 46)^2 + 8800$ (in units of bikes). Suppose that in a certain month, the remaining number of shared bikes reaches its maximum. Determine whether this maximum exceeds the parking capacity at that time.

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