
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
(1) Given $m=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\cos x-x+3\sin 2x)dx$, find the constant term in the expansion of $\left(x-\frac{1}{2\sqrt{x}}\right)^{3m}$.

(2) Compute $\int_{-1}^{1}(\sqrt{1-x^2}-1)dx=$ \_\_\_\_\_\_\_.

(3) Let $n$ be a positive integer, and let $f(n)=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$. It is given that $f(2)=\frac{3}{2}$, $f(4)>2$, $f(8)>\frac{5}{2}$, and $f(16)>3$. Based on these results, estimate the value of $f(4096)>$ \_\_\_\_\_\_\_.

(4) If the complex number $z=(x+i)(1+i)$ is purely imaginary, where $x$ is a real number and $i$ is the imaginary unit, find the conjugate of $z$, denoted as $\overline{z}=$ \_\_\_\_\_\_\_.

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