
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 function $f(x) = e^x + ae^{-x} - 2x$ is an odd function.  
(Ⅰ) Find the value of the real number $a$, and determine the monotonicity of $f(x)$.  
(Ⅱ) Let the function $g(x) = f(2x) - 4bf(x)$, when $x > 0$, $g(x) > 0$ always holds, find the range of the real number $b$.

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