
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 the function $f(x)=a^{x+1}-2$ $(a > 0$ and $a\neq 1)$. If the graph of $y=f(x)$ passes through the point $(1,7)$.

$(1)$ Find the value of $a$ and the zero of $y=f(x)$.

$(2)$ Find the solution set of the inequality $f(x) \geqslant - \frac{5}{3}$.

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