
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]


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Water is the source of life and one of the indispensable important material resources for human survival and development. In order to better manage water quality and protect the environment, the Municipal Sewage Treatment Office plans to purchase 10 sewage treatment equipment in advance. There are two models, $A$ and $B$, with their prices and sewage treatment capacities as shown in the table below:<br/>

|          | $A$ model | $B$ model |
|----------|-----------|-----------|
| Price (million yuan) | $12$ | $10$ |
| Sewage treatment capacity (tons/month) | $240$ | $200$ |

$(1)$ In order to save expenses, the Municipal Sewage Treatment Office plans to purchase sewage treatment equipment with a budget not exceeding $105$ million yuan. How many purchasing plans do you think are possible?<br/>
$(2)$ Under the condition in $(1)$, if the monthly sewage treatment volume must not be less than $2040$ tons, to save money, please help the Municipal Sewage Treatment Office choose the most cost-effective plan.

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