
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 farmer contracted several acres of fruit trees. This year, the farmer invested $13,500$ yuan and harvested a total fruit yield of $19,000$ kilograms. There are two sales methods to choose from: one is to sell directly in the orchard, and the other is to sell in the market. The selling price per kilogram is $4$ yuan in the orchard and $x$ yuan in the market $\left(x \gt 4\right)$. When selling in the market, an average of $1,000$ kilograms can be sold per day. (Profit $=$ Revenue $-$ Cost)<br/>$(1)$ If all these fruits are sold in the market, it will take ______ days;<br/>$(2)$ The income from selling all in the market is ______ yuan more than selling all directly in the orchard (expressed in an algebraic expression containing $x$);<br/>$(3)$ The farmer sells $6,000$ kilograms directly in the orchard and the rest in the market. Express the total profit of the farmer this year in an algebraic expression containing $x$ and simplify it.

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