
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 a set M consisting of all functions $f(x)$ that satisfy the following conditions: (i) the equation $f(x) - x = 0$ has a real root; (ii) let $f'(x)$ be the derivative of $f(x)$, and for any $x$ within the domain of $f(x)$, $f'(x) > 1$.
(Ⅰ) Determine whether the function $f(x) = 2x + \sin x$ is an element of set M, and explain why.
(Ⅱ) If the function $g(x) = \ln x + ax$ is an element of set M, determine the range for the real number $a$.

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