
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
Let $f(x) = \log_2(1 + a \cdot 2^x + 4^x)$, where $a$ is a constant.  
(1) Find the value of $a$ when $f(2) = f(1) + 2$;  
(2) Determine the range of values for $a$ such that for $x \in [1, +\infty)$, the inequality $f(x) \geq x - 1$ always holds.

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