
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 $f(x) = ax^2 + 2x + c$ (where $a, c \in \mathbb{N}^*$), it satisfies the following conditions:
1. $f(1) = 5$;
2. $6 < f(2) < 11$.
   - Find the values of $a$ and $c$;
   - If for any real number $x$, the inequality $f(x) - 2mx \leq 1$ holds, determine the range of values for the real number $m$.

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