
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 certain community in our city is planning to build a new nursing home this year, with a total of 100 rooms planned for three types of nursing rooms. These three types of nursing rooms are single rooms (1 nursing bed), double rooms (2 nursing beds), and triple rooms (3 nursing beds). Due to actual needs, the number of single rooms is between 10 and 30 (including 10 and 30), and the number of double rooms is twice the number of single rooms. Let $t$ be the number of single rooms planned to be built.

(I) According to the question, fill in the table below:

| Number of Single Rooms | 10 | ... | $t$ | ... | 30 |
|------------------------|----|-----|-----|-----|----|
| Number of Double Rooms | ___ | ... | $2t$ | ... | 60 |
| Number of Triple Rooms | 70 | ... | ___ | ... | ___ |
| Number of Nursing Beds | 260 | ... | ___ | ... | ___ |

(II) If the nursing home can provide 200 nursing beds after completion, find the value of $t$.

(III) Find the maximum and minimum number of nursing beds that the nursing home can provide after completion.

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