
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 functions $f(x) = x^2 - ax$ and $g(x) = mx + n\ln x$, the slope of the tangent line to the graph of $f(x)$ at point $(1, f(1))$ is 1, and the function $g(x)$ achieves a local minimum value of $2 - 2\ln 2$ at $x=2$.
(1) Find the expressions for functions $f(x)$ and $g(x)$.
(2) Knowing that the inequality $f(x) + g(x) \geqslant x^2 - \lambda(x - 1)$ holds for any $x \in (0, 1]$, find the range of the real number $\lambda$.

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