
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 novice economist-cryptographer received a cryptogram from the ruler, which contained a new secret decree regarding the introduction of a per-unit tax on a certain market. The cryptogram indicated the total tax revenue that needed to be collected, also emphasizing that a greater sum of tax revenues could not be collected on this market. Unfortunately, the economist-cryptographer decoded the cryptogram incorrectly, misordering the digits in the total tax revenue. Based on the incorrect data, a decision was made to introduce a per-unit tax on the producer amounting to 90 monetary units per unit of goods. It is known that the market demand is given by \( Q_d = 688 - 4P \), and the market supply is linear. In a situation with no taxes, at the equilibrium point, the price elasticity of market supply is 1.5 times the modulus of the price elasticity of the market demand function. After the tax was introduced, the producer's price decreased to 64 monetary units.

1. Restore the market supply function.
2. Determine the total tax revenue that was collected with the chosen tax rate.
3. Determine the per-unit tax rate that would achieve the ruler's decree.
4. What is the amount of tax revenue that the ruler indicated to collect?

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