
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 shop uses a balance with two unequal arms to weigh gold. The ratio of the lengths of the left and right arms is $\lambda$. A customer comes to the shop to buy $10$ grams of gold. The shopkeeper first places a $5g$ weight on the left pan of the balance and takes out some gold to put on the right pan to balance the scale. Then the shopkeeper places the $5g$ weight on the right pan of the balance and takes out some gold to put on the left pan to balance the scale. Finally, the gold weighed twice is given to the customer.  
$(1)$ Analyze whether the customer receives less than $10g$, exactly $10g$, or more than $10g$ of gold. Explain why.  
$(2)$ If the shopkeeper places a $10g$ weight on the left pan of the balance and then takes out some gold to put on the right pan to balance the scale, to minimize the sum of the gold weighed three times, what should the shop set the ratio of the lengths of the left and right arms, $\lambda$, to be? Explain the reason.

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