
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 an old wall that is 14m long. Now, it plans to use this old wall to build a rectangular workshop with an area of 126 m<sup>2</sup>. The engineering conditions are as follows: ① The cost of building 1m of new wall is a yuan; ② The cost of repairing 1m of old wall is $$\frac {a}{4}$$ yuan; ③ The cost of demolishing 1m of old wall and using the materials to build 1m of new wall is $$\frac {a}{2}$$ yuan; ④ The cost for the roof and the floor is b yuan. After discussion, there are two plans:  
(1) Use a section of the old wall, x m (x < 14), as one side of the rectangular workshop;  
(2) The rectangular workshop uses the full length of the old wall as one side, with x (x ≥ 14). The question is, how should the old wall be utilized, that is, when x is how many meters, will the construction cost be the lowest?

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