
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 enterprise plans to invest a total of $200$ million yuan in two projects, A and B, with each project requiring a minimum investment of $10$ million yuan. Based on preliminary market research, the relationship between the profit $p(t)$ (in million yuan) of Project A and the investment amount $t$ (in million yuan) is given by $p(t)=at^{3}+21t$; the relationship between the profit $g(t)$ (in million yuan) of Project B and the investment amount $t$ (in million yuan) is given by $g(t)=-2a(t-b)^{2}$ (where $b<200$). If the investment in Project A is $x$ million yuan, the total profit of the two projects is denoted by $f(x)$ (in million yuan). It is known that when $30$ million yuan is invested in Project A, the profit of Project A is $180$ million yuan, and the profit of Project B is $120$ million yuan. Find:<br/>$(1)$ The analytical expression for $f(x)$.<br/>$(2)$ How to allocate the investment amounts for Projects A and B to maximize the total profit $f(x)$, and determine the maximum value of $f(x)$.

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