
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 that investing $x$ ten-thousand yuan in commodity A yields a profit of $P = \frac{x}{4}$ and investing $x$ ten-thousand yuan in commodity B yields a profit of $Q = \frac{a}{2}\sqrt{x}$ ($a>0$), find the minimum value of $a$ such that investing 20 ten-thousand yuan to sell both or just one of these commodities yields a profit that is not less than 5 ten-thousand yuan.

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