
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 a function $f(x)$ defined on $\mathbb{R}$ with a period of 3, and for $1 \leq x \leq 3$, $f(x) = x^2 + 4$.

(1) Find the value of $f(5) + f(7)$;  
(2) If the equation $f(x) = mx^2$ ($m \in \mathbb{R}$) has real roots in the interval $[4, 6]$, find the range of the real number $m$.

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