
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 $f(x)= \begin{cases} cx+1 \quad (0 < x < c) \\ 2^{- \frac {x}{c^{2}}}+1 \quad (c\leqslant x < 1)\\ \end{cases}$ 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}
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