
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 a function $f(x) = a + \frac{1}{4^x + 1}$ with its domain as $\mathbb{R}$ is an odd function.  
(1) Find the value of $a$ and describe the monotonicity of the function $f(x)$.  
(2) If for any $t \in \mathbb{R}$, the inequality $f(t^2 - 2t) + f(2t^2 - k) < 0$ always holds, find the range of values for $k$.

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