
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 the function $f(x) = |x - 3|$.

$(1)$ If $f(t) + f(2t) < 9$, find the range of $t$.

$(2)$ If there exists $x \in [2, 4]$ such that $f(2x) + |x + a| \leqslant 3$ holds, find the range of $a$.

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