
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
For a function $f(x)$, if there exists a real number $m$ such that $f(x+m)-f(m)$ is an odd function over $\mathbb{R}$, then $f(x)$ is called a "shifted odd function" with shift difference $m$.  
(1) Determine whether the functions $f(x) = 2x + 1$ and $g(x) = 2^x$ are shifted odd functions, and explain the reasons;  
(2) If $f(x) = \sin(x+\varphi)$ is a shifted odd function with shift difference $\frac{\pi}{4}$, find the value of $\varphi$;  
(3) If $f(x) = x^3 + bx^2 + cx$ is not a shifted odd function for any $m$ belonging to the interval $[-\frac{1}{2}, +\infty)$, find the conditions that the real numbers $b$ and $c$ must satisfy.

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