
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
Low-carbon living has become a trend in today's society, and people's environmental awareness is gradually deepening. "Low-carbon environmental protection, green travel" has become everyone's life philosophy, and many people choose to travel by bicycle. A certain company sells two models of bicycles, model A and model B. The purchase price of model A is $500$ yuan per unit, and the purchase price of model B is $800$ yuan per unit. The company can make a profit of $650$ yuan by selling $3$ model A bicycles and $2$ model B bicycles, and a profit of $350$ yuan by selling $1$ model A bicycle and $2$ model B bicycles.  
$(1)$ What is the profit for the company when selling one model A bicycle and one model B bicycle?  
$(2)$ To meet the public demand, the company plans to purchase a total of $20$ model A and model B bicycles, with a budget not exceeding $13000$ yuan. How many model A bicycles at least need to be purchased?

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