
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 $f(x) = x^3 - 9x^2\cos\alpha + 48x\cos\beta + 18\sin^2\alpha$, $g(x) = f'(x)$, and for any real number $t$, it holds that $g(1+e^{-|t|}) \geq 0$, $g(3+\sin t) \leq 0$.

(1) Find the value of $\cos\alpha + 2\cos\beta$.

(2) If $\varphi(x) = \frac{1}{3}x^3 - 2x^2\cos\beta + x\cos\alpha$, and let $h(x) = \ln\varphi(x)$, for any $x \in [0, 1]$, the inequality $h(x+1-m) < h(2x+2)$ always holds, find the range of the real number $m$.

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