
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 $f(x)=ax^{2}+2x+c$ ($a, c \in \mathbb{N}^{*}$) satisfies: $(1) f(1)=5$; $(2) 6 < f(2) < 11$.  
$(1)$ Find the values of $a$ and $c$;  
$(2)$ If for any real number $x \in \left[ \frac{1}{2}, \frac{3}{2} \right]$, the inequality $f(x)-2mx \leqslant 1$ holds, find the range of the real number $m$.

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