
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 $f(x) = x - 1 - a \ln x$.

(1) If $a=2$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(4, f(4))$.

(2) Discuss the minimum value of $f(x)$ for $a \in \mathbb{R}$ and find the value of $a$ such that $f(x) \geqslant 0$.

(3) Let $m$ be an integer, and for any positive integer $n$, $(1+\frac{1}{2})(1+\frac{1}{2^{2}}) \ldots (1+\frac{1}{2^{n}}) < m$. Find the minimum value of $m$.

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