
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
Chongqing is known as the "infrastructure maniac" city. In February of this year, Chongqing's rail transit achieved a "new breakthrough in operating mileage exceeding 500 kilometers", and other rail projects in Chongqing are also under construction.  
$(1)$ The original plan for the construction mileage in the first quarter of this year (including construction of ordinary roads, elevated roads, and tunnels) was a total of 56 kilometers, with 32 kilometers for ordinary road construction. The length of elevated road construction is at least 7 times the length of tunnel construction. What is the maximum length of tunnel construction in the first quarter of this year?  
$(2)$ If the construction mileage in the first quarter is completed exactly according to the original plan and the tunnel construction mileage reaches the maximum value, it is known that the cost per kilometer of ordinary road construction, elevated road construction, and tunnel construction in the first quarter is 1 billion yuan, 2 billion yuan, and 4 billion yuan, respectively. In the second quarter, the total mileage is expected to decrease by 10a kilometers, the tunnel construction mileage will increase by a kilometers, and the elevated road construction will decrease by 2a kilometers. The cost per kilometer of ordinary road construction and tunnel construction in the second quarter is the same as in the first quarter, while the cost per kilometer of elevated road construction will increase by 0.5a billion yuan. If the total cost in the second quarter is the same as in the first quarter, find the value of a.

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