
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 recent years, the emergence of shared bicycles has provided great convenience for citizens' "green travel". A shared bicycle company plans to invest a total of $2$ million yuan in two cities, City A and City B. Each city must invest at least $0.7$ million yuan. According to the preliminary market research, the profit $P$ (in units of million yuan) in City A is related to the investment $a$ (in units of million yuan) by $P = 2\sqrt{5a} - 8$, and the profit $Q$ in City B is related to the investment $a$ by $Q = \frac{1}{4}a + 3$. <br/>$(1)$ When investing $1.25$ million yuan in City A, find the total profit of the company; <br/>$(2)$ How should the investments in City A and City B be arranged to maximize the total profit?

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