
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
Given the equation \(a^{x} + 2x - 4 = 0 \) \((a > 0, a \neq 1)\) has roots \(u_{1}, u_{2}, \ldots, u_{k}\), where \(k\) is a positive integer, and the equation \(\log_{a}{ (2x) } + x - 2 = 0 \) \((a > 0, a \neq 1)\) has roots \(v_{1}, v_{2}, \ldots, v_{l}\), where \(l\) is a positive integer, find the value of \(\frac{u_{1} + u_{2} + \cdots + u_{k} + v_{1} + v_{2} + \cdots + v_{l}}{k + l}\).

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