
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 $f(x)=\begin{cases} cx+1, & 0 < x < c, \\ 2^{-\frac{x}{c^2}}+1, & c\leqslant x < 1 \end{cases}$ satisfies $f(c^2)=\frac{9}{8}$.

$(1)$ Find the value of the constant $c$;

$(2)$ Solve the inequality $f(x) > \frac{\sqrt{2}}{8}+1$.

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