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

(1) Find the value of the real number $c$;

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

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