
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
Initially, 1500 yuan is allocated to purchase $x$ units of product A and $y$ units of product B. However, the price of product A increases by 1.5 yuan per unit and the price of product B increases by 1 yuan per unit. Despite reducing the purchase quantity of product A by 10 units, the total cost still exceeds the original budget by 29 yuan. Additionally, if the price of both products increases by only 1 yuan per unit and the purchase quantity of product A is reduced by 5 units from the initial plan, the total cost for both products amounts to 1563.5 yuan.
(1) Determine the relationship between $x$ and $y$.
(2) If the sum of twice the initially planned quantity of product A and the initially planned quantity of product B is greater than 205 but less than 210, find the originally planned purchase quantities of products A and B.

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