
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 $f(x) = 4^{x} - 2 \cdot 2^{x+1} + a$, where $x \in [0,3]$.
- $(1)$ If the minimum value of $f(x)$ is $1$, find the value of $a$.
- $(2)$ If there exists $x \in [0,3]$ such that $f(x) \geq 33$, find the range of values for $a$.

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