
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
A certain shopping mall plans to invest a sum of money to purchase a batch of goods. After market research, two sales methods are found:<br/>Method 1: If sold at the end of the month, a profit of $30\%$ can be made, but there is a storage cost of $900$ yuan;<br/>Method 2: If sold at the beginning of the month, a profit of $20\%$ can be made, and the principal and profit can be reinvested in other goods, resulting in an additional profit of $5\%$ by the end of the month.<br/>If the shopping mall invests a principal of $x$ yuan, answer the following questions:<br/>$(1)$ The profit obtained by selling at the end of the month is ______ yuan (expressed in the simplest algebraic expression containing $x$), and the profit obtained by selling at the beginning of the month is ______ yuan (expressed in the simplest algebraic expression containing $x$);<br/>$(2)$ If the shopping mall invests a principal of $20000$ yuan, which sales method should be chosen to earn more profit? How much profit is earned at this time?

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