
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
$17.$ Given that $a$ and $b$ are positive real numbers satisfying $a + b = 1$.

$(1)$ Let $t = ab$, find the range of $a$ and $t$, respectively.

$(2)$ Find the minimum value of $y = (a + \frac{1}{a})(b + \frac{1}{b})$.

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