
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 factory has a 14m long old wall and plans to use this wall to construct a rectangular building with an area of 126m<sup>2</sup>. The construction constraints are as follows:
1. The cost to build 1m of new wall is a yuan.
2. The cost to repair 1m of old wall is $$\frac{a}{4}$$ yuan.
3. The cost to remove 1m of old wall and use the obtained materials to build 1m of a new wall is $$\frac{a}{2}$$ yuan.

There are two plans to discuss:
(1) Use a length of x m (0 < x < 14) of the old wall as one side of the rectangle.
(2) The rectangle's side length that uses the old wall is x≥14.
Which plan is more cost-effective for utilizing the old wall to build the factory?

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